US20160224515A1

US20160224515A1 - Method and system for solving the lagrangian dual of a constrained binary quadratic programming problem using a quantum annealer

Info

Publication number: US20160224515A1
Application number: US15/014,576
Authority: US
Inventors: Pooya RONAGH; Ehsan Iranmanesh; Brad Woods
Original assignee: 1QB Information Technologies Inc
Current assignee: 1QB Information Technologies Inc
Priority date: 2015-02-03
Filing date: 2016-02-03
Publication date: 2016-08-04
Also published as: US11514134B2; US20230222173A1; CA2881033C; US20200272683A1; CA2881033A1

Abstract

A method is disclosed for solving the Lagrangian dual of a constrained binary quadratic programming problem. The method comprises obtaining a constrained quadratic binary programming problem; until a convergence is detected, iteratively, performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem, providing the unconstrained quadratic binary programming problem to a quantum annealer, obtaining from the quantum annealer at least one corresponding solution, using the at least one corresponding solution to generate a new approximation for the Lagrangian dual bound; and providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem after convergence.

Description

CROSS-REFERENCE TO RELATED APPLICATION

The present patent application claims priority on Canadian Patent Application No. 2,881,033, filed on Feb. 3, 2015.

FIELD

The invention relates to computing. More precisely, this invention pertains to a method and system for solving the Lagrangian dual problem corresponding to a binary quadratic programming problem.

BACKGROUND

Duality is an important phenomenon in optimization theory. In general, duality is a process of generating a “dual” problem for the original “primal” problem. Solving dual problems provide information about the primal problem.
In some optimization models, duality directly yields alternative viewpoints to the problems. For example, in models of electrical networks, the “primal variables” may represent current flows, whereas the “dual variables” may represent voltage differences. In models of economic markets, “primal variables” may represent production and consumption levels and the “dual variables” may represent prices of goods and service. (See “Applied Lagrange Duality for Constrained Optimization” by Robert M. Freund, 2004, Massachusetts Institute of Technology).
In a wider range of scenarios, dual problems may yield more complicated information about the optimization model. For example, a good dual solution can be used to bound the values of the primal solutions. Using such information may be less trivial however very beneficial to solving optimization problems. For example, the solution to a dual problem can be used to prove optimality of a primal solution. In yet more complicated applications, solving a series of dual sub-problems can be used in iterative fashion to solve a more complicated original primal problem. Examples of such iterative applications of duality include branch and bound methods, cut and bound methods, and decomposition methods for solving integer and mixed-integer optimization problems. (see “Nonlinear Integer Programming” by Duan Li and Xiaoling Sun) (See “Deterministic Methods for Mixed Integer Nonlinear Programming” by Sven Leyffer, PhD Thesis, 1993, University of Dundee). In duality theory, several different types of dualization and dual problems are proposed. One type of dual problems are the Lagrangian dual problems. A thorough description of Lagrangian duality theory is disclosed in “Nonlinear integer programming” by Duan Li and Xiaoling Sun. Lagrangian dual problems can be used to solve many integer programming problems such as non-linear knapsack problems (see “Knapsack Problems” by Hans Kellerer, Ultich Pferschy and David Pisinger), non-linear minimum spanning tree problems (see “The quadratic minimum spanning tree problem” by Arjang Assad and Weixuan Xu), etc. These combinatorial optimization problems are models of many problems of interest in operational research; e.g. scheduling problems, job-shop problems, and resource allocation problems. For applications of Lagrangian techniques in discrete optimization refer to “A survey of Lagrangean techniques for discrete optimization” by Jeremy F. Shapiro, Operations Research Center, Massachusetts Institute of Technology, Cambridge, Mass. and to “Lagrangean relaxation for integer programming” by A. M. Geoffrion, Mathematics Programming Study 2 (1974) 82-114, North-Holland Publishing Company.
There are several methods proposed for solving the Lagrangian dual problems, e.g. subgradient method, outer Lagrangian linearization method, and bundle method. (see Chapter 3 of “Nonlinear integer programming” by Duan Li and Xiaoling Sun). The difficulty of having efficient implementations of such algorithms is the urge to very efficient methods for solving hard nonlinear integer programming problems in various stages of these methods. For example, the single constrained quadratic 0-1 knapsack problem can be solved using an efficient branch and bound method based on Lagrangian duality as explained in Section 11.5, “Nonlinear integer programming” by Duan Li and Xiaoling Sun but the proposed method cannot be generalized for multi-dimensional knapsack problems.
There is therefore a need for a method for solving the Lagrangian dual optimization problems that will overcome the above-identified drawback.
Features of the invention will be apparent from review of the disclosure, drawings and description of the invention below.

BRIEF SUMMARY

According to a broad aspect, there is disclosed a method for solving the Lagrangian dual of a constrained binary quadratic programming problem, the method comprising use of a processor for obtaining a constrained quadratic binary programming problem; until a convergence is detected, use of a processor for iteratively performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem; providing the unconstrained quadratic binary programming problem to a quantum annealer; obtaining from the quantum annealer at least one corresponding solution; using the at least one corresponding solution to generate a new approximation for a Lagrangian dual bound and use of a processor for providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem after the convergence.
In accordance with an embodiment, the use of a processor for obtaining of a constrained quadratic binary programming problem comprises use of a processor for obtaining data representative of an objective function ƒ(x) having a degree less than or equal to two; use of a processor for obtaining data representative of equality constraints having a degree less than or equal to two; and use of a processor for obtaining data representative of inequality constraints having a degree less than or equal to two.
In accordance with an embodiment, the use of a processor for obtaining the constrained quadratic binary programming problem comprises use of a processor for obtaining the constrained quadratic binary programming problem from at least one of a user, a computer, a software package and an intelligent agent.
In accordance with an embodiment, the use of a processor for obtaining of the constrained quadratic binary programming problem further comprises use of a processor for initializing software parameters and use of a processor for initializing a linear programming procedure.
In accordance with an embodiment, the software parameters are obtained by the processor from at least one of a user, a computer, a software package and an intelligent agent.
In accordance with an embodiment, the use of a processor for initializing of the software parameters comprises use of a processor for providing an embedding of the constrained quadratic binary programming problem on the quantum annealer; use of a processor for providing an embedding solver function for providing a list of solutions; use of a processor for providing one of lower and upper bounds and default values for Lagrange multipliers; use of a processor for providing one of initial values and default values for Lagrange multipliers; and use of a processor for providing an error tolerance value for duality gap.
In accordance with an embodiment, the linear programming procedure is carried out until the convergence is detected.
In accordance with an embodiment, the using of the at least one corresponding solution to generate a new approximation for the Lagrangian dual bound comprises using the at least one corresponding solution in the linear programming procedure.
In accordance with an embodiment, the use of a processor for providing of a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem comprises storing the corresponding solution to a file.
In accordance with a broad aspect, there is disclosed a digital computer comprising a central processing unit; a display device; a communication port for operatively connecting the digital computer to a quantum annealer; a memory unit comprising an application for solving the Lagrangian dual of a constrained binary quadratic problem, the application comprising: instructions for obtaining a constrained binary quadratic problem; instructions for iteratively performing a Lagrangian relaxation of the constrained quadratic problem to provide an unconstrained quadratic programming problem; instructions for providing the unconstrained quadratic programming problem to the quantum annealer using the communication port; instructions for obtaining from the quantum annealer via the communication port at least one corresponding solution and for using the at least one corresponding solution to generate a new approximation for a Lagrangian dual bound; instructions for providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem once a convergence is detected and a data bus for interconnecting the central processing unit, the display device, the communication port and the memory unit.
In accordance with a broad aspect, there is disclosed a non-transitory computer-readable storage medium for storing computer-executable instructions which, when executed, cause a digital computer to perform a method for solving the Lagrangian dual of a constrained binary quadratic programming problem, the method comprising obtaining a constrained quadratic binary programming problem; until a convergence is detected, iteratively: performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem; providing the unconstrained quadratic binary programming problem to a quantum annealer; obtaining from the quantum annealer at least one corresponding solution; using the at least one corresponding solution to generate a new approximation for a Lagrangian dual bound and providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem causing the convergence.
In accordance with an embodiment, there is disclosed a use of the method disclosed herein for solving a maximum weighted k-clique problem.
According to a broad aspect, there is disclosed a method for solving the Lagrangian dual of a constrained binary quadratic programming problem, the method comprising obtaining a constrained quadratic binary programming problem; until a convergence is detected, iteratively, performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem, providing the unconstrained quadratic binary programming problem to a quantum annealer, obtaining from the quantum annealer at least one corresponding solution, using the at least one corresponding solution to generate a new approximation for a Lagrangian dual bound; and providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem after the convergence.
An advantage of the method disclosed herein is that it provides a method for using Lagrangian duality in various applications, for example finding Lagrangian based bounds to integer and mixed-integer programming problems using a quantum annealer.
It will be further appreciated that the method disclosed herein greatly improves the processing of a system for solving a Lagrangian dual of a constrained binary quadratic programming problem which is of great advantage.

DESCRIPTION OF THE DRAWINGS

In order that the invention may be readily understood, embodiments of the invention are illustrated by way of example in the accompanying drawings.

FIG. 1 is a flowchart that shows an embodiment of a method for solving the Lagrangian dual of a constrained binary quadratic programming problem using a quantum annealer.

FIG. 2 is a diagram of an embodiment of a system in which the method for solving the Lagrangian dual of a constrained binary quadratic programming problem using a quantum annealer may be implemented. In this embodiment, the system comprises a digital computer and a quantum annealer.

FIG. 3 is a diagram that shows an embodiment of a digital computer used in the system for solving the Lagrangian dual of a constrained binary quadratic programming problem using a quantum annealer.

FIG. 4 is a flowchart that shows an embodiment for providing a constrained binary programming problem.

FIG. 5 is a flowchart that shows an embodiment for initializing software parameters used in an embodiment of the method for solving the Lagrangian dual of a constrained binary quadratic programming problem.

FIG. 6 is a flowchart that shows an embodiment for interpreting the at least one solution provided by the quantum annealer as new cuts for a linear programming procedure in progress in order to determine a new upper approximation for the Lagrangian dual bound of a constrained binary quadratic programming problem.

Further details of the invention and its advantages will be apparent from the detailed description included below.

DETAILED DESCRIPTION

In the following description of the embodiments, references to the accompanying drawings are by way of illustration of an example by which the invention may be practiced.

Terms

The term “invention” and the like mean “the one or more inventions disclosed in this application,” unless expressly specified otherwise.
The terms “an aspect,” “an embodiment,” “embodiment,” “embodiments,” “the embodiment,” “the embodiments,” “one or more embodiments,” “some embodiments,” “certain embodiments,” “one embodiment,” “another embodiment” and the like mean “one or more (but not all) embodiments of the disclosed invention(s),” unless expressly specified otherwise.
A reference to “another embodiment” or “another aspect” in describing an embodiment does not imply that the referenced embodiment is mutually exclusive with another embodiment (e.g., an embodiment described before the referenced embodiment), unless expressly specified otherwise.
The terms “including,” “comprising” and variations thereof mean “including but not limited to,” unless expressly specified otherwise.
The terms “a,” “an”, “the” and “at least one” mean “one or more,” unless expressly specified otherwise.
The term “plurality” means “two or more,” unless expressly specified otherwise.
The term “herein” means “in the present application, including anything which may be incorporated by reference,” unless expressly specified otherwise.
The term “e.g.” and like terms mean “for example,” and thus does not limit the term or phrase it explains. For example, in a sentence “the computer sends data (e.g., instructions, a data structure) over the Internet,” the term “e.g.” explains that “instructions” are an example of “data” that the computer may send over the Internet, and also explains that “a data structure” is an example of “data” that the computer may send over the Internet. However, both “instructions” and “a data structure” are merely examples of “data,” and other things besides “instructions” and “a data structure” can be “data.”
The term “i.e.” and like terms mean “that is,” and thus limits the term or phrase it explains. For example, in the sentence “the computer sends data (i.e., instructions) over the Internet,” the term “i.e.” explains that “instructions” are the “data” that the computer sends over the Internet.
The term “constrained binary quadratic programming” problem and like terms mean finding the minimum of a quadratic real polynomial y=ƒ(x) in several binary variables x=(x₁, . . . , x_n) subject to a (possibly empty) family of equality constraints determined by a (possibly empty) family of m equations g_j(x)=0 for j=1, . . . , m and a (possibly empty) family of inequality constraints determined by a (possibly empty) family of
inequalities h_j(x)≦0 for j=1, . . . ,
Here all functions g_iand h_jare polynomials of degree at most two.
$\min f (x)$ $subject to$ $_{i} (x) = 0 \forall i \in {1, \dots, m}$ $h_{j} (x) \leq 0 \forall j \in {1, \dots, l}$ $x_{k} \in {0, 1} \forall k \in {1, \dots, n}$
The above constrained binary quadratic programming problem will be denoted by (P) and the optimal value of it will be denoted by v(P). An optimal solution x, i.e. a vector at which the objective function attains the value v(P) will be denoted by x*.
It will be appreciated that any quadratic polynomial y=q(x) can be represented in matrix notation y=q(x)=x^tA x+B^tx+C where the matrix A is a real symmetric positive semi-definite square matrix of size n, B is a real vector of size n, and C is a real number.
It will be further appreciated that according to the equality x_i ²=x_i(i=1, . . . , n) for the binary variables, it can be assumed that the function q(x) is given without linear terms; that is, q(x)=x^tQ x+b, where the matrix Q is a real symmetric square matrix of size n, and b is a real constant.
The term “unconstrained binary quadratic programming” problem and like terms mean finding a minimum of an objective function y=x^tQ x+b where Q is a symmetric square real matrix of size n, and b is a real number, also known as the bias of the objective function. The domain of the function is all vectors x ε Bⁿ={0, 1}ⁿwith binary entries.
The term “Lagrangian relaxation” of the constrained binary quadratic programming problem (P), corresponding to fixed Lagrange multipliers λ ε
^mand μ ε
means solving the following optimization problem:
$\min_{x \in {0, 1}^{n}} (f (x) + \sum_{i = 1}^{m} λ_{i} _{i} (x) + \sum_{j = 1}^{l} μ_{j} h_{j} (x))$
The value of the above optimization is denoted as δ_p(λ, μ) and is known to be a lower bound for the optimal value of the original constrained binary quadratic programming, that is, δ_p(λ, μ)≦v(P).
The term “Lagrangian dual” of a constrained binary quadratic programming problem, is used for the following optimization problem:
$\max_{λ, μ} \min_{x} (f (x) + \sum_{i = 1}^{m} λ_{i} _{i} (x) + \sum_{j = 1}^{l} μ_{j} h_{j} (x))$ $subject to$ $x \in {0, 1}^{n}$ $λ \in ℝ^{m}$ $μ \in ℝ_{\geq 0}^{l}$
The value of the above optimization is denoted by δ(P) and is known to be a lower bound for the optimal value of the original constrained binary quadratic programming, that is, δ(P)≦v(P). This value is unique and is also called the “Lagrangian dual bound” for the original constrained binary quadratic programming problem.
The term “optimal Lagrange multiplier,” and the like, will refer to a, not necessarily unique, set of optimal points λ* and μ* at which the value δ(P) is attained for the above optimization problem.
The term “solution to the Lagrangian dual problem” of an original constrained binary quadratic programming problem, refers to the following collection of information received after solving the Lagrangian dual problem: (1) the (unique) optimal value of the Lagrangian dual problem, also known as the Lagrangian dual bound; (2) a set of (not necessarily unique) optimal Lagrange multipliers as described above; and (3) a set of (non necessarily unique) binary vectors at which the optimal value of the Lagrangian dual problem is obtained at the given optimal Lagrange multipliers.
One widely studied class of constrained binary quadratic programming problems is that of linearly constrained ones. In this case the functions g_iand h_jare all linear. Hence the problem can be rewritten as
$\min f (x)$ $subject to$ $A_{eq} x = b_{eq}$ $A_{ineq} x \leq b_{ineq}$ $x_{i} \in {0, 1} \forall i \in {1, \dots, n}$
where y=ƒ(x) is a quadratic polynomial of several binary variables x=(x₁, . . . , x_n) subject to a (possibly empty) family of linear equality constraints determined by a linear system A_eqx=b_eqwhere A_eqis a matrix of size m×n and b_eqis a column matrix of size m×1 and a (possibly empty) family of inequality constraints determined by A_ineqx≦b_ineqwhere A_ineqis a matrix of size l×n and b_ineqis a column matrix of size l×1. A Lagrangian relaxation of the above problem can be written as
$\min_{x} f (x) + λ^{t} (A_{eq} x - b) + μ^{t} (A_{ineq} x - b_{ineq})$ $subject to$ $x_{i} \in {0, 1}, \forall i \in {1, \dots, n}$
for given Lagrange multipliers λ and μ and the Lagrangian dual can be stated as
$\max_{λ, μ} \min_{x} (f (x) + λ^{t} (A_{eq} x - b) + μ^{t} (A_{ineq} x - b_{ineq}))$ $subject to$ $x_{i} \in {0, 1}, \forall i \in {1, \dots, n}$ $λ \in ℝ^{m}$ $μ \in ℝ_{\geq 0}^{l} .$
The term “quantum annealer” and like terms mean a system consisting of one or many types of hardware that can find optimal or sub-optimal solutions to an unconstrained binary quadratic programming problem. An example of this is a system consisting of a digital computer embedding a binary quadratic programming problem as an Ising spin model, attached to an analog computer that carries optimization of a configuration of spins in an Ising spin model using quantum annealing as described, for example, in Farhi, E. et al., “Quantum Adiabatic Evolution Algorithms versus Simulated Annealing” arXiv.org:quant-ph/0201031 (2002), pp 1-16. An embodiment of such analog computer is disclosed by McGeoch, Catherine C. and Cong Wang, (2013), “Experimental Evaluation of an Adiabiatic Quantum System for Combinatorial Optimization” Computing Frontiers.” May 14-16, 2013 (http://www.cs.amherst.edu/ccm/cf14-mcgeoch.pdf) and also disclosed in the patent application US2006/0225165. It will be appreciated that the “quantum annealer” may also be comprised of “classical components,” such as a classical computer. Accordingly, a “quantum annealer” may be entirely analog or an analog-classical hybrid.
The term “embedding” of a binary optimization problem, and the like, refer to an assignment of a set of the quantum bits {q_i1, q_i2, . . . , q_il _i} to each binary variable x_i. Specifications of the role of such an embedding in solving an unconstrained binary quadratic programming problem and presentation of an efficient algorithm for it are disclosed for instance in “A practical heuristic for finding graph minors”—Jun Cai, William G. Macready, Aidan Roy, in U.S. patent application US 2008/0218519 and in U.S. Pat. No. 8,655,828 B2.
The term “embedding solver,” and the like, refer to a function, procedure, and algorithm that consist of instructions for receiving an unconstrained binary quadratic programming problem, carrying a query to the quantum annealer using a provided embedding, and returning at least one result, each result containing a vector of binary entries, representative of a binary point in the domain of the provided unconstrained binary quadratic programming, the value of the objective function of unconstrained binary quadratic programming at that point, and the number of occurrences of the result in the entire number of reads.
The term “callback function,” and the like, refer to a user function that is called iteratively by the software throughout the run time. In the system disclosed herein, there is only one callback function which determines how the queries to the quantum annealer are carried. In other words, the embedding solver explained above is provided by the user as a callback function.
Neither the Title nor the Abstract is to be taken as limiting in any way as the scope of the disclosed invention(s). The title of the present application and headings of sections provided in the present application are for convenience only, and are not to be taken as limiting the disclosure in any way.
Numerous embodiments are described in the present application, and are presented for illustrative purposes only. The described embodiments are not, and are not intended to be, limiting in any sense. The presently disclosed invention(s) are widely applicable to numerous embodiments, as is readily apparent from the disclosure. One of ordinary skill in the art will recognize that the disclosed invention(s) may be practiced with various modifications and alterations, such as structural and logical modifications. Although particular features of the disclosed invention(s) may be described with reference to one or more particular embodiments and/or drawings, it should be understood that such features are not limited to usage in the one or more particular embodiments or drawings with reference to which they are described, unless expressly specified otherwise.
It will be appreciated that the invention may be implemented in numerous ways, including as a method, a system, a computer readable medium such as a computer readable storage medium. In this specification, these implementations, or any other form that the invention may take, may be referred to as systems or techniques. A component such as a processor or a memory described as being configured to perform a task includes both a general component that is temporarily configured to perform the task at a given time or a specific component that is manufactured to perform the task.
With all this in mind, the present invention is directed to a method and system for solving the Lagrangian dual of a constrained binary quadratic programming problem.
Now referring to FIG. 2, there is shown an embodiment of a system 200 in which an embodiment of the method for solving the Lagrangian dual of a constrained binary quadratic programming problem may be implemented.
The system 200 comprises a digital computer 202 and a quantum annealer 204.
The digital computer 202 receives a constrained binary quadratic programming problem and provides a solution to the Lagrangian dual of the constrained binary quadratic programming problem.
It will be appreciated that the constrained binary quadratic programming problem may be provided according to various embodiments.
In one embodiment, the constrained binary quadratic programming problem is provided by a user interacting with the digital computer 202.
Alternatively, the constrained binary quadratic programming problem is provided by another computer, not shown, operatively connected to the digital computer 202. Alternatively, the constrained binary quadratic programming problem is provided by an independent software package. Alternatively, the constrained binary quadratic programming problem is provided by an intelligent agent.
Similarly, it will be appreciated that the solution to the Lagrangian dual of the constrained binary quadratic programming problem may be provided according to various embodiments.
In accordance with an embodiment, the solution to the Lagrangian dual of the constrained binary quadratic programming problem is provided to the user interacting with the digital computer 202.
Alternatively, the solution to the Lagrangian dual of the constrained binary quadratic programming problem is provided to another computer operatively connected to the digital computer 202.
In fact, it will be appreciated by the skilled addressee that the digital computer 202 may be any type of computer.
In one embodiment, the digital computer 202 is selected from a group consisting of desktop computers, laptop computers, tablet PCs, servers, smartphones, etc.
Now referring to FIG. 3, there is shown an embodiment of a digital computer 202. It will be appreciated that the digital computer 202 may also be broadly referred to as a processor.
In this embodiment, the digital computer 202 comprises a central processing unit (CPU) 302, also referred to as a microprocessor or a processor, a display device 304, input devices 306, communication ports 308, a data bus 310 and a memory unit 312.
The CPU 302 is used for processing computer instructions. The skilled addressee will appreciate that various embodiments of the CPU 302 may be provided.
In one embodiment, the central processing unit 302 is a CPU Core i5-3210M running at 2.5 GHz and manufactured by Intel™.
The display device 304 is used for displaying data to a user. The skilled addressee will appreciate that various types of display device 304 may be used.
In one embodiment, the display device 304 is a standard liquid-crystal display (LCD) monitor.
The communication ports 308 are used for sharing data with the digital computer 202.
The communication ports 308 may comprise, for instance, a universal serial bus (USB) port for connecting a keyboard and a mouse to the digital computer 202.
The communication ports 308 may further comprise a data network communication port such as an IEEE 802.3 port for enabling a connection of the digital computer 202 with another computer via a data network.
The skilled addressee will appreciate that various alternative embodiments of the communication ports 308 may be provided.
In one embodiment, the communication ports 308 comprise an Ethernet port.
The memory unit 312 is used for storing computer-executable instructions.
It will be appreciated that the memory unit 312 comprises, in one embodiment, an operating system module 314.
It will be appreciated by the skilled addressee that the operating system module 314 may be of various types.
In an embodiment, the operating system module 314 is OS X Yosemite (Version 10.10.5) manufactured by Apple™.
The memory unit 312 further comprises an application for solving the Lagrangian dual of a constrained binary quadratic programming problem 316.
The application 316 comprises instructions for obtaining a constrained quadratic binary programming problem.
The application 316 further comprises instructions for iteratively performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem; providing the unconstrained quadratic binary programming problem to a quantum annealer; obtaining from the quantum annealer at least one corresponding solution and for using the at least one corresponding solution to generate a new approximation for the Lagrangian dual bound.
The application 316 further comprises instructions for providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem once a convergence is detected.
Each of the central processing unit 302, the display device 304, the input devices 306, the communication ports 308 and the memory unit 312 is interconnected via the data bus 310.
Now referring back to FIG. 2, it will be appreciated that the quantum annealer 204 is operatively connected to the digital computer 202.
It will be appreciated that the coupling of the quantum annealer 204 to the digital computer 202 may be achieved according to various embodiments.
In one embodiment, the coupling of the quantum annealer 204 to the digital computer 202 is achieved via a data network.
It will be appreciated that the quantum annealer 204 may be of various types.
In one embodiment, the quantum annealer 204 is manufactured by D-Wave Systems Inc. More information on this embodiment of a quantum annealer applicable to 204 may be found at http://www.dwavesys.com. The skilled addressee will appreciate that various alternative embodiments may be provided for the quantum annealer 204.
More precisely, the quantum annealer 204 receives an unconstrained binary quadratic programming problem from the digital computer 202.
The quantum annealer 204 is capable of solving the unconstrained binary quadratic programming problem and of providing at least one corresponding solution. In the case where a plurality of corresponding solutions is provided, the plurality of corresponding solutions may comprise optimal and suboptimal solutions.
The at least one corresponding solution is provided by the quantum annealer 204 to the digital computer 202.
Now referring to FIG. 1 and according to processing step 102, a constrained binary quadratic programming problem is provided. It will be appreciated that in one embodiment the processor is used for providing the constrained binary quadratic programming problem.
Now referring to FIG. 4, there is shown an embodiment for providing a constrained binary quadratic programming problem.
As mentioned above, the constrained binary quadratic programming problem can be referred to as:
$\min f (x)$ $subject to$ $_{i} (x) = 0 \forall i \in {1, \dots, m}$ $h_{j} (x) \leq 0 \forall j \in {1, \dots, l}$ $x_{k} \in {0, 1} \forall k \in {1, \dots, n}$
According to processing step 402, data representative of an objective function ƒ(x) are provided. It will be appreciated that in one embodiment the processor is used for providing the data representative of an objective function ƒ(x).
According to processing step 404, data representative of the equality constraints are provided. It will be appreciated that in one embodiment the processor is used for providing the data representative of the equality constraints.
According to processing step 406, data representative of the inequality constraints are provided. It will be appreciated that in one embodiment the processor is used for providing the data representative of the inequality constraints.
It will be appreciated that the providing of a constrained binary quadratic programming problem may be performed according to various embodiments.
As mentioned above and in one embodiment, the constrained binary quadratic programming problem is provided by a user interacting with the digital computer 202. Alternatively, the constrained binary quadratic programming problem is provided by another computer operatively connected to the digital computer 202. Alternatively, the constrained binary quadratic programming problem is provided by an independent software package. Alternatively, the constrained binary quadratic programming problem is provided by an intelligent agent.
Now referring to FIG. 1 and according to processing step 104, parameters of the software are initialized. It will be appreciated that in one embodiment the processor is used for initializing the parameters of the software.
Now referring to FIG. 5, there is shown an embodiment for initializing parameters or using default values for them.
In one embodiment, the software parameters are obtained by the processor from at least one of a user, a computer, a software package and an intelligent agent.
According to processing step 502, data representative of an embedding of the constrained binary quadratic programming problem on the quantum annealer are provided. In one embodiment the processor is used for providing the data representative of an embedding of the constrained binary quadratic programming problem on the quantum annealer. The embedding should be such that it respects all logical couplings between variables as they occur in the functions ƒ(x), g₁(x), . . . , g_m(x) and h₁(x), . . . , h_l(x). In other words the chains of qubits corresponding to two variable x_rand x_sshould have a coupling between them whenever there is a nonzero coefficient for the term x_rx_sin at least one of the functions ƒ(x), g₁(x), . . . , g_m(x) and h₁(x), . . . , h_l(x).
In one embodiment, the data comprise an array of n sets, with each entry being a set {q_i1, . . . , q_il _i} of quantum bits of the annealer. Still in one embodiment, the data of the embedding of the constrained binary quadratic programming problem are stored in the reserved name embeddings, in the namespace ORACLE. Hence in ORACLE::embeddings.
Still referring to FIG. 5 and according to processing step 504, an embedding solver function is provided as a callback function. It will be appreciated that in one embodiment the processor is used for providing an embedding solver function. In one embodiment, the function is implemented by the user in the namespace ORACLE, as ORACLE::solve_qubo.
The input parameter of the callback function is a pointer to an instance of the data type ORACLE::qubo, representative of an unconstrained binary quadratic programming neglecting the corresponding bias of it.
The output of the call back function is a pointer to an instance of the type ORACLE::result, representative of a list of optimal and suboptimal solutions to the unconstrained binary quadratic programming problem.
The following is an example of a code snippet in C++ for providing the callback function using the API developed by DWave:


#include “dwave_sapi.h”
#include <ostream>
#include <string.h>
ORACLE::result* ORACLE::solve_qubo(ORACLE::qubo& qubo) {
sapi_Solver* embedding_solver = NULL;
char err_msg[SAPI_ERROR_MAX_SIZE];
sapi_Connection* connection = NULL;
const char* url = “https://.../”;
const char* token = “...”;
const char* system = “...”;
sapi_remoteConnection(url, token, NULL, &connection, err_msg);
sapi_getSolver(connection, system, &solver, err_msg);
sapi_SolveParameters params;
params.len =1;
params.elements= new (sapi_SolveParameterEntry*) [1];
params.elements[0].name= “num_reads”;
stringstream num_reads_string << 100 * qubo.dim;
params.elements[0].value= num_reads_string.str( )).c_str( ) ;
sapi_embeddingSolver(solver,*ORACLE::embeddings,
embedding_solver, err_msg);
QUBO::result* result = NULL;
sapi_solveQubo(embedding_solver, qubo, params,
&result,err_msg);
return qubo_result;
}

It will be appreciated that, in one embodiment, the providing of the unconstrained binary optimization problem to the quantum annealer is achieved using a processor.
More precisely, it will be appreciated that in one embodiment a token system is used over the Internet to provide access to the quantum annealer remotely and to authenticate use.
It will be appreciated that in one embodiment the at least one solution is provided in a table by the quantum annealer, according to the instructions of use of the quantum annealer. In one embodiment, the DWave system, provides these solutions in the data type sapi_IsingResult* which is then type-casted automatically to an instance of QUBO::result*.
Still referring to FIG. 5 and according to processing step 506, lower and upper bounds for Lagrange multipliers are provided. It will be appreciated that in one embodiment the lower and upper bounds for Lagrange multipliers are provided using the processor.
It will be appreciated that in one embodiment the providing of these real numbers of type double is achieved by overwriting the names ORACLE::dual_lb and ORACLE::dual_ub. Each of these types will be required to contain an array of doubles of size m+l. The first m entries of these arrays represent, respectively, the lower and upper bounds for the Lagrange multipliers corresponding to the m equality constraints and the last l of them represent, respectively, the lower and upper bounds for the Lagrange multipliers corresponding to the l inequality constraints.
It will be appreciated that if these names are not overwritten, the values of them are initialized with the default values.
In the mathematical formulae below, the vector of lower bounds is denoted by lb=(lb₁, . . . , lb_m+l) and the vector of upper bounds is denoted by ub=(ub₁, . . . , ub_m+l).
In one embodiment, the default lower bound for a Lagrange multiplier corresponding to an equality constraint is −1e6 and the default upper bound for it is +1e6.
The default lower bound for a Lagrange multiplier corresponding to an inequality constraint is 0 and the default upper bound for it is +1e6.
Still referring to FIG. 5 and according to processing step 508, initial values for the Lagrange multipliers are provided. It will be appreciated that in one embodiment the initial values for the Lagrange multipliers are provided using the processor.
In fact, it will be appreciated that the providing of these real valued numbers are achieved by overwriting the name ORACLE::dual_init_val with an array of doubles of size m+l. If this name is not overwritten, the values are initialized with default values.
The default initial value for any of the Lagrange multipliers, corresponding to any of equality or inequality constraints, is 0.
In the mathematical formulae below, the vector of all initial values of Lagrange multipliers is denoted by init=(init₁, . . . , init_m+l).
According to processing step 510, an error tolerance value for equality and inequality conditions is provided in the software. It will be appreciated that in one embodiment the error tolerance value for equality and inequality conditions is provided using the processor.
Unless overwritten by the user and according to one embodiment, the error tolerance value is initialized to 1e-5 and stored as ORACLE::tol. The error tolerance value is used in several points in the software.
The Lagrangian dual bound and the value of the best feasible solution are considered equal if their difference is less that ORACLE::tol, in which case strong duality is assumed to hold.
More generally, any system of linear inequalities LHS≦RHS is considered satisfied if the value of all entries in LHS−RHS is at most ORACLE::tol.
Similarly, any system of linear equalities LHS=RHS is considered satisfied if the absolute value of all entries in LHS−RHS is at most ORACLE::tol.
Now referring back to FIG. 1 and according to processing step 106 a linear programming procedure is initialized.
It will be appreciated that the linear programming procedure will be carried iteratively during the entire runtime of the method and will terminate once the result of it converges; i.e., it does not improve in the two consecutive iterations.
In first iteration, the linear programming problem, referred to as (L), is the following:
$\max y$ $subject to$ $y \in (- \infty, + \infty)$ $λ_{i} \in (- l b_{i}, {ub}_{i}) \forall i \in {1, \dots, m}$ $μ_{j} \in (- l b_{j + m}, {ub}_{j + m}) \forall j \in {1, \dots, }$
At this point, the value of y is set to y*=+∞ and the variables λ_iand μ_iplay no role in the objective function or in the constraints. The vectors λ=(λ₁, . . . , λ_m) and μ=(μ₁, . . . , μ_l) will represent the Lagrange multipliers in the linear programming problem (L) formulated above.
It will be appreciated that the values of the variables λ_iand μ_jmay be set to any arbitrary feasible ones. λ*_i=init_i(∀i=1, . . . , m) and μ*_j=init_j+m(∀j=1, . . . , m).
It will be appreciated that in further processing steps a number of inequality constraints will be added to the linear programming problem (L) and a simplex-based linear programming method proceeds to find new optimal values for y*, λ* and μ* such that they maximize y and satisfy all added constraints.
According to processing step 108, the Lagrangian relaxation of the constrained binary quadratic programming problem corresponding to the Lagrange multipliers λ* and μ* are provided.
As mentioned above, the Lagrangian relaxation of the constrained binary quadratic programming problem corresponding to the Lagrange multipliers λ* and μ* is the following optimization problem:
$\min_{x \in {0, 1}^{n}} (f (x) + \sum_{i = 1}^{m} λ_{i}^{*} _{i} (x) + \sum_{j = 1}^{l} μ_{j}^{*} h_{j} (x))$
It will be appreciated that this optimization problem is an unconstrained binary quadratic programming problem and therefore can be completely determined by a symmetric real matrix Q of size n and a bias constant b, representative of an objective function of the form x^tQ x+b in n binary variables. It will be appreciated that the unconstrained binary quadratic programming problem is provided to the quantum annealer. As mentioned above, the unconstrained binary quadratic problem may be provided to the quantum annealer according to various embodiments.
In one embodiment, the information of this unconstrained binary quadratic programming problem is stored in a variable Q of type ORACLE::qubo forgetting the bias term.
Still referring to FIG. 1 and according to processing step 110, the call back function ORACLE::solve_qubo is now called with input Q to provide at least one corresponding solution for the unconstrained binary quadratic programming problem from the quantum annealer.
It will be appreciated that the at least one corresponding solution of the unconstrained binary quadratic programming is achieved with a pointer to an instance of type ORACLE::result.
Now referring to processing step 112, each of the at least one corresponding solution provided according to processing step 110, is used to generate a new linear constraint (a.k.a. cut) for the linear programming problem (L).
Now referring to FIG. 6, there is shown an embodiment for interpreting the at least one corresponding solution as new cuts for the linear programming procedure in progress and for determining a new approximation for the Lagrangian dual bound.
According to processing step 602, at least one corresponding solution is provided. In the case where the at least one corresponding solution comprises a plurality of solutions, the plurality of solutions S comprises optimal as well as suboptimal solutions as mentioned above.
According to processing step 604, a new cut corresponding to each solution is determined using the solution itself and the current Lagrange multipliers.
In fact and given a binary vector s of size n, it will be appreciated that the following inequality cuts out a half-space of the space R^m×R_≧ ^l×R, of the Lagrange multipliers and the variable y.
y≦f(s)+Σ_i=1 ^mλ*_i g _i(s)+Σ_j=1 ^lμ*_j h _j(s)
According to processing step 606, all the new cuts corresponding to all vectors s ε S are added to the linear programming procedure (L) in progress.
Using a simplex-based linear programming method, the previous solution to the linear programming problem (L) is now modified to a solution to the modified version of the linear programming problem (L) formulated below:
$\max y$ $subject to all previous constraints$ $and y \leq f (s) + \sum_{i = 1}^{m} λ_{i}^{*} g_{i} (s) + \sum_{j = 1}^{} μ_{j}^{*} h_{j} (s), \forall s \in S$
According to processing step 608, the optimal value of y* and the optimal multipliers λ* and μ* are updated by solving the modified linear programming problem (L), and y* is now recognized as an upper approximation for the Lagrangian dual bound.
Referring to FIG. 1 and according to processing step 114, a test is performed in order to find out if the optimal value of y* is or not improved from a previous value.
In the case where the optimal value of y* does not improve and according to processing step 116, the results of the optimization are reported. It will be appreciated that in one embodiment, the results of the optimization are provided using the processor.
It will be appreciated that in one embodiment, the output of this method comprises the value of y* representative of the optimal value, and δ(P) of the Lagrangian dual problem corresponding to the constrained binary quadratic programming problem formulated above.
The output of this method further comprises the optimal multipliers λ* and μ*, representative of a set of Lagrange multipliers at which the value δ(P) is attained by the Lagrangian dual of the constrained binary quadratic programming problem.
Still in this embodiment, the output of the method further comprises the minimum value encountered for ƒ(s) among all solutions s generated by the quantum annealer and provided by the callback function ORACLE::solve_qubo.
Finally, the output comprises an indicator of whether δ(P) and the minimum value encountered for ƒ(s) are at most of distance ORACLE::tol from each other.
In one embodiment, the above information is stored using the digital computer in a file.
It will be appreciated that a non-transitory computer-readable storage medium is further disclosed. The non-transitory computer-readable storage medium is used for storing computer-executable instructions which, when executed, cause a digital computer to perform a method for solving the Lagrangian dual of a constrained binary quadratic programming problem, the method comprising obtaining a constrained quadratic binary programming problem; until a convergence is detected, iteratively, performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem, providing the unconstrained quadratic binary programming problem to a quantum annealer, obtaining from the quantum annealer at least one corresponding solution, using the at least one corresponding solution to generate a new approximation for the Lagrangian dual bound; and providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem after the convergence.
It will be appreciated that an advantage of the method disclosed herein is that it enables an efficient method for finding the Lagrangian dual bound for a constrained binary quadratic programming problem using a quantum annealer.
It will be further appreciated that the method disclosed herein improves the processing of a system for solving the Lagrangian dual of a constrained binary quadratic programming problem.
It will be appreciated that the method disclosed herein may be used for solving various problems.
For instance, the method disclosed herein may be used for solving the maximum weighted k-clique problem. The maximum weighted k-clique problem may be formulated as:
$\max x^{t} A x$ $subject to \sum_{i = 1}^{n} x_{i} = k$ $x_{i} \in {0, 1} \forall i \in {1, \dots, n}$
In this embodiment, A is a symmetric square matrix of size n representing the weights of edges of a graph with n vertices. The binary variable x_irepresents selection of the vertex labelled by positive integer i ε {1, . . . , n}.
It is appreciated that the mentioned maximization problem may be written as the minimization of the negative of the objective function:
$\min - x^{t} Ax$ $subject to \sum_{i = 1}^{n} x_{i} = k$ $x_{i} \in {0, 1} \forall i \in {1, \dots, n}$
In one embodiment, let a graph with 5 vertices, represents a group of 5 coworkers. To each pair of coworkers, a utility factor is assigned for the collaboration between the two coworkers. The utilities can be represented with an upper triangular matrix:
$W = (\begin{matrix} 0 & 0 & 3 & 5 & 2 \\ 0 & 0 & 1 & - 1 & 4 \\ 0 & 0 & 0 & 7 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix})$
For example, the utility of collaboration of person 3 with person 5 is 3.
The utility matrix W may as well be represented by the 5×5 symmetric matrix A=1/2(W+W^t).
$A = (\begin{matrix} 0 & 0 & 1.5 & 2.5 & 1 \\ 0 & 0 & 0.5 & - 0.5 & 2 \\ 1.5 & 0.5 & 0 & 3.5 & 1.5 \\ 2.5 & - 0.5 & 3.5 & 0 & 0 \\ 1 & 2 & 1.5 & 0 & 0 \end{matrix})$
The problem of selecting the most productive team of 3 people amongst the 5 coworkers is then an instance of the maximum weighted 3-clique problem which is denoted by (C):
$\min - x^{t} Ax$ $subject to \sum_{i = 1}^{5} x_{i} - 3 = 0$ $x_{i}, \dots, x_{5} \in {0, 1}$
The single constraint of (C), indicates that there is only a single Lagrange multiplier, denoted by λ, present in any Lagrangian relaxation of it. The lower and upper bounds −10 and 10 are provided for the Lagrange multiplier with initial value 0.
A linear programming problem is then initiated as:
$\max y$ $subject to y \in (- \infty, + \infty)$ $λ \in (- 10, 10)$
which initially has +∞ as optimal solution.
At the initial Lagrange multiplier λ=0 the unconstrained quadratic binary programming problem:
$\min_{x \in {0, 1}^{n}} - x^{t} Ax$
is solved by a quantum annealer resulting the optimal value −24 obtained at the binary vector (1, 1, 1, 1, 1).
The linear programming problem is then modified to:
$\max y$ $y \leq - 24 + 2 λ$ $λ \in (- 10, 10)$
A simplex-based linear programming method gives an optimal value of −4 obtained at λ=10 for this linear programming problem.
At the new Lagrangian multiplier λ=10, the unconstrained quadratic binary programming problem:
$\min_{x \in {0, 1}^{n}} - x^{t} Ax + 10 (\sum_{i = 1}^{5} x - 3)$
is solved by a quantum annealer resulting the optimal value −30 obtained at the binary vector (0, 0, 0, 0, 0).
The linear programming problem is then modified to
$\max y$ $subject to y \leq - 24 + 2 λ$ $y \leq - 3 λ$ $λ \in (- 10, 10)$
A simplex-based linear programming method gives an optimal value of −30 obtained at λ=4.8 for this linear programming problem.
At the new Lagrangian multiplier λ=4.8, the unconstrained quadratic binary programming problem:
$\min_{x \in {0, 1}^{n}} - x^{t} Ax + 4.8 (\sum_{i = 1}^{5} x - 3)$
is solved by a quantum annealer resulting the optimal value −15.2 obtained at the binary vector (1, 0, 1, 1, 1).
The linear programming problem is then modified to:
$\max y$ $subject to y \leq - 24 + 2 λ$ $y \leq - 3 λ$ $y \leq - 20 + λ$ $λ \in (- 10, 10)$
A simplex-based linear programming method gives an optimal value of −15 obtained at λ=5 for this linear programming problem.
Finally, at the new Lagrangian multiplier λ=5, the unconstrained quadratic binary programming problem
$\min_{x \in {0, 1}^{n}} - x^{t} Ax + 5 (\sum_{i = 1}^{5} x - 3)$
is solved by a quantum annealer resulting the optimal value −15 obtained at the binary vector (1, 0, 1, 1, 0).
$\max y$ $subject to y \leq - 24 + 2 λ$ $y \leq - 3 λ$ $y \leq - 20 + λ$ $y \leq - 15$ $λ \in (- 10, 10)$
A simplex-based linear programming method gives an optimal value of −15 obtained at λ=5 for this linear programming problem.
Since the optimal value of the linear program, has not improved from the previous iteration, convergence has occurred.
The best known feasible solution is (1, 0, 1, 1, 0) and the value of the objective function of (C) at this point is −15.
Since the optimal value of the linear program is also −15 strong duality has occurred. The output of the method is (1) the Lagrangian dual bound −15, (2) the feasible binary vector (1, 0, 1, 1, 0) and (3) a flag indicating strong duality has occurred.
For the application at hand, the solution to the method disclosed is interpreted as the selection of a team of 3 people, consisting of person 1, 3 and 4 as the most productive team of size 3, amongst the 5 coworkers.
The skilled addressee will appreciate that the method disclosed herein is therefore of great advantage for solving this problem. In fact, the method disclosed greatly improves the processing of a system used for solving such problem.
Although the above description relates to specific embodiments as presently contemplated by the inventors, it will be understood that the invention in its broad aspect includes functional equivalents of the elements described herein.

Claims

1. A method for solving the Lagrangian dual of a constrained binary quadratic programming problem, the method comprising:

use of a processor for obtaining a constrained quadratic binary programming problem;

until a convergence is detected, use of a processor for iteratively:

performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem;

providing the unconstrained quadratic binary programming problem to a quantum annealer;

obtaining from the quantum annealer at least one corresponding solution;

using the at least one corresponding solution to generate a new approximation for a Lagrangian dual bound;

use of a processor for providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem after the convergence.

2. The method as claimed in claim 1, wherein the use of a processor for obtaining of a constrained quadratic binary programming problem comprises:

use of a processor for obtaining data representative of an objective function ƒ(x) having a degree less than or equal to two;

use of a processor for obtaining data representative of equality constraints having a degree less than or equal to two; and

use of a processor for obtaining data representative of inequality constraints having a degree less than or equal to two.

3. The method as claimed in claim 1, wherein the use of a processor for obtaining the constrained quadratic binary programming problem comprises use of a processor for obtaining the constrained quadratic binary programming problem from at least one of a user, a computer, a software package and an intelligent agent.

4. The method as claimed in claim 1, wherein the use of a processor for obtaining of the constrained quadratic binary programming problem further comprises use of a processor for initializing software parameters and use of a processor for initializing a linear programming procedure.

5. The method as claimed in claim 4, wherein the software parameters are obtained by the processor from at least one of a user, a computer, a software package and an intelligent agent.

6. The method as claimed in claim 4, wherein the use of a processor for initializing of the software parameters comprises:

use of a processor for providing an embedding of the constrained quadratic binary programming problem on the quantum annealer;

use of a processor for providing an embedding solver function for providing a list of solutions;

use of a processor for providing one of lower and upper bounds and default values for Lagrange multipliers;

use of a processor for providing one of initial values and default values for Lagrange multipliers; and

use of a processor for providing an error tolerance value for duality gap.

7. The method as claimed in claim 4, wherein the linear programming procedure is carried out until the convergence is detected.

8. The method as claimed in claim 7, wherein the using of the at least one corresponding solution to generate a new approximation for the Lagrangian dual bound comprises using the at least one corresponding solution in the linear programming procedure.

9. The method as claimed in claim 1, wherein the use of a processor for providing of a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem comprises storing the corresponding solution to a file.

10. A digital computer comprising:

a central processing unit;

a display device;

a communication port for operatively connecting the digital computer to a quantum annealer;

a memory unit comprising an application for solving the Lagrangian dual of a constrained binary quadratic problem, the application comprising:

instructions for obtaining a constrained binary quadratic problem;

instructions for iteratively performing a Lagrangian relaxation of the constrained quadratic problem to provide an unconstrained quadratic programming problem;

instructions for providing the unconstrained quadratic programming problem to the quantum annealer using the communication port;

instructions for obtaining from the quantum annealer via the communication port at least one corresponding solution and for using the at least one corresponding solution to generate a new approximation for a Lagrangian dual bound;

instructions for providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem once a convergence is detected; and

a data bus for interconnecting the central processing unit, the display device, the communication port and the memory unit.

11. A non-transitory computer-readable storage medium for storing computer-executable instructions which, when executed, cause a digital computer to perform a method for solving the Lagrangian dual of a constrained binary quadratic programming problem, the method comprising obtaining a constrained quadratic binary programming problem; until a convergence is detected, iteratively: performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem; providing the unconstrained quadratic binary programming problem to a quantum annealer; obtaining from the quantum annealer at least one corresponding solution; using the at least one corresponding solution to generate a new approximation for a Lagrangian dual bound and providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem causing the convergence.

12. Use of the method claimed in claim 1 for solving a maximum weighted k-clique problem.