arXiv subjects

Shu-Cherng Fang

Publications and source records attributed to Shu-Cherng Fang.

7 recordsLinked to original sources

arXiv2026

Variable Aggregation-based Perspective Reformulation for Mixed-Integer Convex Optimization with Symmetry

This paper addresses the challenging issue of symmetry in mixed-integer convex optimization problems, which frequently arise in real-world applications such as the unit commitment problem. Although variable aggregation techniques have been employed to mitigate symmetry, their impact on tightening the corresponding continuous relaxation has not been thoroughly investigated. In this work, we propose a new formulation that integrates the perspective reformulation method into the variable aggregation framework, yielding a tighter continuous relaxation for mixed-integer convex optimization problems with symmetric structures. We prove that, in the presence of symmetry, the convex hull of the feasible region associated with each set of aggregated variables can be exactly characterized. These results demonstrate the effectiveness of the proposed reformulation and establish new theoretical foundations for achieving tightness in variable aggregation-based mixed-integer programming formulations.

math.OC↗

arXiv2025

An Efficient Solution Method for Solving Convex Separable Quadratic Optimization Problems

Convex separable quadratic optimization problems occur in many practical applications. In this paper, based on an iterative resolution scheme of the KKT system, we develop an efficient method for solving a quadratic programming problem with a convex separable objective function subject to multiple convex separable constraints. We show that the proposed approach leads to a dual coordinate ascent algorithm and provide a convergence proof. Numerical experiments support the superior performance of the proposed method to that of the Gurobi solver, especially for solving large-scale convex separate quadratic programming problems.

math.OC↗

arXiv2025

Enhancing Quadratic Programming Solvers via Quadratic Nonconvex Reformulation

In this paper, we consider solving nonconvex quadratic programming problems using modern solvers such as Gurobi and SCIP. It is well-known that the classical techniques of quadratic convex reformulation can improve the computational efficiency of global solvers for mixed-integer quadratic optimization problems. In contrast, the use of quadratic nonconvex reformulation (QNR) has not been previously explored. This paper introduces a QNR framework--an unconventional yet highly effective approach for improving the performance of state-of-the-art quadratic programming solvers such as Gurobi and SCIP. Our computational experiments on diverse nonconvex quadratic programming problem instances demonstrate that QNR can substantially accelerate both Gurobi and SCIP. Notably, with QNR, Gurobi achieves state-of-the-art performance on several benchmark and randomly generated instances.

math.OC↗

arXiv2023

Semidefinite Programming Approximation for a Matrix Optimization Problem over an Uncertain Linear System

A matrix optimization problem over an uncertain linear system on finite horizon (abbreviated as MOPUL) is studied, in which the uncertain transition matrix is regarded as a decision variable. This problem is in general NP-hard. By using the given reference values of system outputs at each stage, we develop a polynomial-time solvable semidefinite programming (SDP) approximation model for the problem. The upper bound of the cumulative error between reference outputs and the optimal outputs of the approximation model is theoretically analyzed. Two special cases associated with specific applications are considered. The quality of the SDP approximate solutions in terms of feasibility and optimality is also analyzed. Results of numerical experiments are presented to show the influences of perturbed noises at reference outputs and control levels on the performance of SDP approximation.

math.OC↗

arXiv2015

Best $L_1$ approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces

In this article, we study the problem of best $L_1$ approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem into an appropriate framework and prove the unicity of best $L_1$ approximation of Heaviside-type functions in an even-dimensional Chebyshev space under the condition that the dimension of the subspace composed of the even functions is half the dimension of the whole space. We also apply the results to compute best $L_1$ approximations of Heaviside-type functions by polynomials and Hermite polynomial splines with fixed knots.

math.FA↗

arXiv2014

Double Well Potential Function and Its Optimization in The n-dimensional Real Space -- Part I

A special type of multi-variate polynomial of degree 4, called the double well potential function, is studied. When the function is bounded from below, it has a very unique property that two or more local minimum solutions are separated by one local maximum solution, or one saddle point. Our intension in this paper is to categorize all possible configurations of the double well potential functions mathematically. In part I, we begin the study with deriving the double well potential function from a numerical estimation of the generalized Ginzburg-Landau functional. Then, we solve the global minimum solution from the dual side by introducing a geometrically nonlinear measure which is a type of Cauchy-Green strain. We show that the dual of the dual problem is a linearly constrained convex minimization problem, which is mapped equivalently to a portion of the original double well problem subject to additional linear constraints. Numerical examples are provided to illustrate the important features of the problem and the mapping in between.

math.OC↗

arXiv2014

Double Well Potential Function and Its Optimization in the n-dimensional Real Space - Part II

In contrast to taking the dual approach for finding a global minimum solution of a double well potential function, in Part II of the paper, we characterize a local minimizer, local maximizer, and global minimizer directly from the primal side. It is proven that, for a ``nonsingular" double well function, there exists at most one local, but non-global, minimizer and at most one local maximizer. Moreover, when it exists, the local maximizer is ``surrounded" by local minimizers in the sense that the norm of the local maximizer is strictly less than that of any local minimizer. We also establish some necessary and sufficient optimality conditions for the global minimizer, local non-global minimizer and local maximizer by studying a convex secular function over specific intervals. These conditions lead to three algorithms for identifying different types of critical points of a given double well function.

math.OC↗