Quantum ising model construction method for security constrained unit commitment optimization problem

Abstract

This invention discloses a method for constructing a quantum Ising model for safety-constrained unit combinatorial optimization problems, relating to the field of quantum computing. The method includes: constructing a safety-constrained unit combinatorial optimization model and obtaining parameters for a mixed-integer programming problem; using the Benders decomposition method to decompose the mixed-integer programming problem into a main problem and subproblems; substituting the optimal binary solution into the subproblems to obtain new cutting planes and expanding the set of cutting planes; constructing a compact high-dimensional quadratic function to fit the set of cutting planes; solving a semi-definite programming problem to obtain the parameters of the high-dimensional quadratic function; transforming the quadratic unconstrained binary optimization model constructed based on the high-dimensional quadratic function into an Ising model; solving the Ising model to obtain the qubit states and obtaining the optimal binary solution; and substituting the optimal binary solution into the above steps for iterative solving. This invention solves the problem of huge qubit resource consumption in existing technologies, especially in dealing with NP-hard problems with complex constraints and many variables.

Description

Technical Field

[0001] This invention relates to the fields of quantum computing technology and power system optimization, specifically to a method for constructing a quantum Ising model for safety-constrained unit combination optimization problems. Background Technology

[0002] In the field of power system optimization, the safety-constrained unit combinatorial optimization problem is described as a mixed-integer linear programming problem (NP-hard), with the objective of finding the optimal solution from the feasible solution set of the combinatorial problem. By solving the mixed-integer linear programming problem, the optimal decision scheme for practical problems can be obtained.

[0003] Quantum computing offers significant advantages in solving quadratic unconstrained binary optimization problems. The quantum annealing algorithm, based on quantum mechanical effects such as tunneling and superposition, has the potential to improve problem-solving speed and the ability to search for the global optimum. Mixed-integer programming problems can be viewed as problems searching for the minimum energy state; all possible solutions correspond to the energy of a certain state, represented by a 0-1 sequence. Therefore, it is necessary to first transform the problem into an equivalent quadratic unconstrained binary optimization model or the Ising model so that the quantum annealer can handle it. Currently, the main approach is to convert the constraints into the objective function in binary form using penalty functions.

[0004] Research has revealed that in many scenarios where quantum computers are used to solve security-constrained combinatorial optimization problems, constructing the Ising model using penalty functions requires a large number of qubits; for example, the penalty coefficient in each iteration requires tens of bits of binary representation. With existing quantum annealing algorithms, as the number of iterations increases, it consumes a significant amount of valuable qubit computational resources, exceeding the maximum number of qubits that current dedicated quantum computers can utilize, thus hindering the reduction of solution time and cost for mixed-integer programming problems. Furthermore, mixed-integer programming is NP-hard, therefore solving it using classical computers requires substantial time. Summary of the Invention

[0005] To address the aforementioned shortcomings in existing technologies, the quantum Ising model construction method for safety-constrained unit combination optimization problems provided by this invention solves the problem of huge qubit resource consumption when using the penalty function method to construct the Ising model and then using quantum computing (especially quantum annealing) to solve the safety-constrained unit combination optimization problem, especially when dealing with NP-hard problems with complex constraints and many variables.

[0006] To achieve the aforementioned objectives, the present invention employs the following technical solution: a method for constructing a quantum Ising model for safety-constrained unit combination optimization problems, comprising the following steps:

[0007] S1: Construct a safety-constrained unit combination optimization model to obtain the parameters of the mixed integer programming problem;

[0008] S2: In classical computers, the Benders decomposition method is used to decompose mixed integer programming problems into the Benders main problem and Benders subproblems;

[0009] S3: Substitute the optimal binary solution to solve the Benders subproblem, obtain new cutting planes and expand the set of cutting planes;

[0010] S4: Construct a compact set of high-dimensional quadratic function fitting cutting planes;

[0011] S5: Solve the semidefinite programming problem through static robust optimization to obtain the parameters of the high-dimensional quadratic function;

[0012] S6: Transform the quadratic unconstrained binary optimization model based on high-dimensional quadratic functions into the Ising model;

[0013] S7: Using cloud technology and a dedicated quantum computer, the Ising model is solved to obtain the state of the qubits and the optimal binary solution is obtained.

[0014] S8: Substitute the optimal binary solution into step S3 for iterative solution until the error between the optimal value of the subproblem objective function and the optimal value of the objective function of the quadratic unconstrained binary optimization problem converges within a preset range, thus completing the construction of the quantum Ising model for the safety-constrained unit combination optimization problem.

[0015] Furthermore, the mixed-integer programming problem in S1 is:

[0016]

[0017] Among them, and They are binary variables and continuous variables, respectively. binary variable Dimensions and The coefficients of the objective function, , and For the constraint matrix, and The right-hand item, superscript This represents the transpose of a matrix.

[0018] Furthermore, the set of cutting planes in S3 is:

[0019]

[0020] in, For the set of cutting planes, and For the first The coefficients of the cutting plane in the next iteration. As an auxiliary variable, This represents the number of iterations.

[0021] At this point, the mixed-integer programming problem is transformed into:

[0022] .

[0023] Furthermore, in S4, by fitting the optimal solution of the Benders principal problem relaxation problem, and by minimizing the value of the high-dimensional quadratic function at the relaxation optimal solution, a compact high-dimensional quadratic function is constructed as follows:

[0024]

[0025]

[0026]

[0027] in, , and The parameters to be optimized are those of a high-dimensional quadratic function. To find the optimal solution to the relaxed Benders principal problem, The regularization coefficient is . Represents any symbol.

[0028] Furthermore, step S5 includes the following sub-steps:

[0029] S51: Treating the infinite-dimensional constraint in a high-dimensional quadratic function as a minimum problem, and its dual as a maximum problem, the formula is as follows:

[0030]

[0031]

[0032]

[0033] in, and To be related to boundary constraints Related dual variables, For the first In each cutting plane , The set, , In order to be with the first Each cutting plane is related, corresponding to The dual variable vector of the boundary constraints, In order to be with the first Vectors associated with each cutting plane , In order to be with the first A scalar related to the cutting plane ;

[0034] S52: Based on the maximum value problem, the infinite-dimensional constraint is transformed into a semidefinite programming problem using the Schur complement property, as shown in the formula:

[0035]

[0036] .

[0037] Furthermore, step S6 includes the following sub-steps:

[0038] S61: Based on the high-dimensional quadratic function parameters obtained from solving semidefinite programming problems , and The quadratic unconstrained binary optimization model is constructed as follows:

[0039]

[0040] Among them, This is the penalty coefficient;

[0041] S62: Transform the quadratic unconstrained binary optimization model into the Ising model:

[0042]

[0043]

[0044]

[0045]

[0046] in, , , The spin variable takes the value of -1 or 1. , and Given parameters, , and It is a binary variable.

[0047] The beneficial effects of this invention are:

[0048] (1) Efficient use of qubits: By approximating the cut plane set through semi-positive definite programming, the complex cut plane set is directly encoded into the Benders main problem in binary form of penalty function, which significantly reduces the auxiliary qubits required to represent cut constraints and the precision bits required to represent large coefficient penalty terms, making it possible to process larger-scale binary variable problems with finite qubits.

[0049] (2) Avoid solving NP-hard safety-constrained combinatorial optimization problems: Quantum Ising machines can replace the traditional Benders decomposition of safety-constrained combinatorial optimization problems to obtain mixed-integer programming Benders main problems (NP-hard), which may achieve faster iteration speeds on some problem instances, especially when quantum computing is used to accelerate the process. Attached Figure Description

[0050] Figure 1 This is a flowchart of a method for constructing a quantum Ising model for a safety-constrained unit combination optimization problem according to the present invention.

[0051] Figure 2 A schematic diagram for constructing a set of cutting planes for fitting a high-dimensional quadratic function.

[0052] Figure 3 The method of this invention is used to calculate the performance charts of 72 and 288 binary variable cases.

[0053] Figure 4 The diagram illustrates the convergence process of calculating 288 binary variable cases using the method of this invention. Detailed Implementation

[0054] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0055] like Figure 1 As shown, a method for constructing a quantum Ising model for the safety-constrained unit combination optimization problem includes the following steps:

[0056] S1: Construct a safety-constrained unit combination optimization model to obtain the parameters of the mixed integer programming problem;

[0057] S2: In classical computers, the Benders decomposition method is used to decompose mixed integer programming problems into the Benders main problem and Benders subproblems;

[0058] S3: Substitute the optimal binary solution to solve the Benders subproblem, obtain new cutting planes and expand the set of cutting planes;

[0059] S4: Construct a compact set of high-dimensional quadratic function fitting cutting planes;

[0060] S5: Solve the semidefinite programming problem through static robust optimization to obtain the parameters of the high-dimensional quadratic function;

[0061] S6: Transform the quadratic unconstrained binary optimization model based on high-dimensional quadratic functions into the Ising model;

[0062] S7: Using cloud technology and a dedicated quantum computer, the Ising model is solved to obtain the state of the qubits and the optimal binary solution is obtained.

[0063] S8: Substitute the optimal binary solution into step S3 for iterative solution until the error between the optimal value of the subproblem objective function and the optimal value of the quadratic unconstrained binary optimization objective function converges within a preset range, thus completing the construction of the quantum Ising model for the safety-constrained unit combination optimization problem.

[0064] The safety-constrained unit combination optimization model in S1 is shown below, used to construct the mixed-integer programming problem for engineering problems. The parameters of the mixed-integer programming problem to be solved are obtained, including the coefficients of the objective function. and constraint matrix , and and the right end item and .

[0065] The mixed-integer programming problem in S1 is:

[0066]

[0067] in, and They are binary variables and continuous variables, respectively. binary variable Dimensions and The coefficients of the objective function, , and For the constraint matrix, and The right-hand item, superscript This represents the transpose of a matrix.

[0068] In S2, the Benders decomposition method is used to decompose the mixed integer programming problem into the main problem and subproblems, resulting in a set of cutting planes. The set of cutting planes is the set of cutting planes generated by iteratively solving the subproblem by substituting the optimal binary solution into it. At this point, the original mixed integer programming problem can be transformed into the following formula.

[0069] The set of cutting planes in S3 is as follows:

[0070]

[0071] in, For the set of cutting planes, and For the first The coefficients of the cutting plane in the next iteration. As an auxiliary variable, This represents the number of iterations.

[0072] At this point, the mixed-integer programming problem is transformed into:

[0073] .

[0074] In S4, the high-dimensional quadratic function is the curve fitted to the semidefinite programming problem of the cutting plane set. To obtain a compact high-dimensional quadratic function, this invention fits the optimal solution of the relaxed master problem and uses the high-dimensional quadratic function value corresponding to this optimal solution as the optimization objective. To ensure that the high-dimensional quadratic function fits the cutting plane set compactly, it is necessary to ensure that any point of it is greater than the cutting plane set. .

[0075] In step S4, by fitting the optimal solution to the relaxation problem, a compact high-dimensional quadratic function is constructed by minimizing the value of the high-dimensional quadratic function at the optimal relaxation solution.

[0076]

[0077]

[0078]

[0079] in, , and The parameters of a high-dimensional quadratic function are obtained from a semidefinite programming problem. This is the optimal solution to the relaxation principal problem. The regularization coefficient is . Represents any symbol.

[0080] In step S5, to handle the infinite-dimensional constraints of the problem in step S4, static robust optimization can be used to transform the constraints into a semi-positive definite programming problem. First, the infinite-dimensional constraints are treated as a minimization problem, which is then dualized into a maximization problem. Finally, the Shure complement property is used to transform the problem into a semi-positive definite programming problem. The specific process is as follows:

[0081] S5 includes the following sub-steps:

[0082] S51: Treating the infinite-dimensional constraint in a high-dimensional quadratic function as a minimum problem, and its dual as a maximum problem, the formula is as follows:

[0083]

[0084]

[0085]

[0086] in, and To be related to boundary constraints Related dual variables, For the first In each cutting plane , The set, , In order to be with the first Each cutting plane is related, corresponding to The dual variable vector of the boundary constraints, In order to be with the first Vectors associated with each cutting plane , In order to be with the first A scalar related to the cutting plane ;

[0087] S52: Based on the maximum value problem, the infinite-dimensional constraint is transformed into a semidefinite programming problem using the Schur complement property, as shown in the formula:

[0088]

[0089] .

[0090] In S6, to adapt to quantum computers, the quadratic unconstrained binary optimization model based on high-dimensional quadratic functions is converted into the Ising model.

[0091] S6 includes the following sub-steps:

[0092] S61: Based on the high-dimensional quadratic function parameters obtained from solving semidefinite programming problems , and The quadratic unconstrained binary optimization model is constructed as follows:

[0093]

[0094] in, This is the penalty coefficient;

[0095] S62: Transform the quadratic unconstrained binary optimization model into the Ising model:

[0096]

[0097]

[0098]

[0099]

[0100] in, , , The spin variable takes the value of -1 or 1. , and Given parameters, , and It is a binary variable.

[0101] In one embodiment of the present invention, Figure 2 A schematic diagram of the cutting plane for constructing a high-dimensional quadratic function fitting. Figure 3 To calculate the performance of 72 and 288 binary variable cases using the method of this invention, respectively, Figure 4 The convergence process of 288 binary variable cases was calculated using the method of this invention, and it can be seen that the method proposed in this invention has certain advantages.

[0102] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the invention.

Claims

1. A method for constructing a quantum Ising model for security constrained unit commitment optimization problem, characterized in that, Includes the following steps: S1: Construct a safety-constrained unit combination optimization model to obtain the parameters of the mixed integer programming problem; S2: In classical computers, the Benders decomposition method is used to decompose mixed integer programming problems into the Benders main problem and Benders subproblems; S3: Substitute the optimal binary solution to solve the Benders subproblem, obtain new cutting planes and expand the set of cutting planes; S4: Construct a compact set of high-dimensional quadratic function fitting cutting planes; In step S4, the optimal solution for relaxation is obtained by solving the relaxation Benders principal problem. A compact high-dimensional quadratic function is then constructed by minimizing the value of the high-dimensional quadratic function at the optimal solution. in, , and The parameters to be optimized are those of a high-dimensional quadratic function. This is the relaxed optimal solution to Benders' principal problem. The regularization coefficient is . Represents any symbol, superscript To represent the transpose of a matrix, It is a binary variable. and For the first The coefficients of the cutting plane in the next iteration. For the set of cutting planes, This represents the number of iterations. S5: Solve the semidefinite programming problem through static robust optimization to obtain the parameters of the high-dimensional quadratic function; S5 includes the following sub-steps: S51: Treating the infinite-dimensional constraint in a high-dimensional quadratic function as a minimum problem, and its dual as a maximum problem, the formula is as follows: in, and To be related to boundary constraints Related dual variables, For the first In each cutting plane , The set, , In order to be with the first Each cutting plane is related, corresponding to The dual variable vector of the boundary constraints, In order to be with the first Vectors associated with each cutting plane , In order to be with the first A scalar related to the cutting plane ; S52: Based on the maximum value problem, the infinite-dimensional constraint is transformed into a semidefinite programming problem using the Schur complement property, as shown in the formula: S6: Transform the quadratic unconstrained binary optimization model based on high-dimensional quadratic functions into the Ising model; S7: Using cloud technology and a dedicated quantum computer, the Ising model is solved to obtain the state of the qubits and the optimal binary solution is obtained. S8: Substitute the optimal binary solution into step S3 for iterative solution until the error between the optimal value of the subproblem objective function and the optimal value of the objective function of the quadratic unconstrained binary optimization problem converges within a preset range, thus completing the construction of the quantum Ising model for the safety-constrained unit combination optimization problem.

2. The method for constructing the quantum Ising model for safety-constrained unit combination optimization problems according to claim 1, characterized in that, The mixed-integer programming problem in S1 is: in, and They are binary variables and continuous variables, respectively. binary variable Dimensions and The coefficients of the objective function, , and For the constraint matrix, and The right-hand item, superscript This represents the transpose of a matrix.

3. The method for constructing the quantum Ising model for safety-constrained unit combination optimization problems according to claim 2, characterized in that, The set of cutting planes in S3 is as follows: in, For the set of cutting planes, and For the first The coefficients of the cutting plane in the next iteration. As an auxiliary variable, This represents the number of iterations. At this point, the Benders main problem of mixed-integer programming is transformed into: 。 4. The method for constructing the quantum Ising model for safety-constrained unit combination optimization problems according to claim 3, characterized in that, S6 includes the following sub-steps: S61: Based on the high-dimensional quadratic function parameters obtained from solving semidefinite programming problems , and The quadratic unconstrained binary optimization model is constructed as follows: in, This is the penalty coefficient; S62: Transform the quadratic unconstrained binary optimization model into the Ising model: in, , , The spin variable takes the value of -1 or 1. , and Given parameters, , and It is a binary variable.

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