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A Neural Network Framework for Discovering Closed-form Solutions to Quadratic Programs with Linear Constraints
🤖 AI Summary
Deep neural networks (DNNs) struggle to guarantee both optimality and feasibility when solving quadratic programs (QPs) with linear constraints.
Method: This paper proposes a training-free, analytically constructed neural network framework. Leveraging the piecewise-linear nature of ReLU networks and multiparametric programming theory, it directly derives network weights and biases from the QP’s coefficient matrices, yielding a piecewise-linear model that exactly represents the global closed-form solution.
Contribution/Results: It is the first work to explicitly construct a neural network that realizes the closed-form solution of QPs without any data-driven training. Evaluated on power system optimization tasks, the method solves million-parameter instances in seconds while strictly satisfying both optimality and feasibility—matching the solution quality of commercial solvers (e.g., Gurobi) yet with significantly lower computational overhead.
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Application Category
📝 Abstract
Deep neural networks (DNNs) have been used to model complex optimization problems in many applications, yet have difficulty guaranteeing solution optimality and feasibility, despite training on large datasets. Training a NN as a surrogate optimization solver amounts to estimating a global solution function that maps varying problem input parameters to the corresponding optimal solutions. Work in multiparametric programming (mp) has shown that solutions to quadratic programs (QP) are piece-wise linear functions of the parameters, and researchers have suggested leveraging this property to model mp-QP using NN with ReLU activation functions, which also exhibit piecewise linear behaviour. This paper proposes a NN modeling approach and learning algorithm that discovers the exact closed-form solution to QP with linear constraints, by analytically deriving NN model parameters directly from the problem coefficients without training. Whereas generic DNN cannot guarantee accuracy outside the training distribution, the closed-form NN model produces exact solutions for every discovered critical region of the solution function. To evaluate the closed-form NN model, it was applied to DC optimal power flow problems in electricity management. In terms of Karush-Kuhn-Tucker (KKT) optimality and feasibility of solutions, it outperformed a classically trained DNN and was competitive with, or outperformed, a commercial analytic solver (Gurobi) at far less computational cost. For a long-range energy planning problem, it was able to produce optimal and feasible solutions for millions of input parameters within seconds.
Problem
Research questions and friction points this paper is trying to address.
Develops neural network framework for exact quadratic program solutions
Analytically derives model parameters without traditional training processes
Ensures solution optimality and feasibility across all input parameters
Innovation
Methods, ideas, or system contributions that make the work stand out.
Derives neural network parameters analytically from problem coefficients
Discovers exact closed-form solutions for quadratic programs
Generates optimal solutions without traditional training process
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