arXiv subjects

Suhan Zhong

Publications and source records attributed to Suhan Zhong.

18 recordsLinked to original sources

arXiv2026

Moment Relaxations for Data-Driven Wasserstein Distributionally Robust Optimization

We propose moment relaxations for data-driven $p$-Wasserstein distributionally robust optimization ($p$-WDRO) problems that are defined by polynomials. The proposed moment relaxations admit Benders-type decomposition with parallel evaluation of the subgradients using each sample subproblem, which enables efficient solution for larger training sets. We then identify conditions on $p$ and the defining polynomial degrees such that the proposed $k$-th order moment relaxations preserve the asymptotic consistency of the original $p$-WDRO (i.e., the relaxation gap is bounded at most linearly by the Wasserstein radius). In particular, these conditions translate to effective bounds on $k$, which lead to polynomially sized semidefinite optimization formulations that are compatible with existing solvers. Numerical experiments on a box-constrained regression problem and a two-stage production problem are included to demonstrate the scalability and the effectiveness of the proposed moment relaxations.

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arXiv2026

Generalized Nash equilibrium problems with quasi-linear constraints

We study generalized Nash equilibrium problems (GNEPs) such that objectives are polynomial functions, and each player's constraints are linear in their own strategy. For such GNEPs, the KKT sets can be represented as unions of simpler sets by Carathéodory's theorem. We give a convenient representation for KKT sets using partial Lagrange multiplier expressions. This produces a set of branch polynomial optimization problems, which can be efficiently solved by Moment-SOS relaxations. By doing this, we can compute all generalized Nash equilibria or detect their nonexistence. This method may not be very scalable to large scale GNEPs. Numerical experiments are provided to demonstrate the computational efficiency.

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arXiv2026

Distributionally robust shortfall risk portfolio model with moment ambiguity sets

This paper employs shortfall risk to measure portfolio risk. Assume stock returns follow polynomial relations with their influence factors. We propose a moment-based distributionally robust optimization (DRO) shortfall risk portfolio model. For piecewise linear loss functions, we show that this DRO model can be transformed into a tractable linear conic optimization problem with nonnegative polynomial cones. A Moment-SOS relaxation algorithm is proposed to solve the transformed problem. Its finite and asymptotic convergence properties are studied. For optimizers computed from our algorithm, we give convenient conditions verify their global optimality for the original DRO problem. Numerical experiments involving real stock market data are given to show the efficiency of our approach.

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arXiv2026

Sparse Copositive Polynomial Optimization

This paper studies the copositive optimization problem whose objective is a sparse polynomial, with linear constraints over the nonnegative orthant. We propose sparse Moment-SOS relaxations to solve it. Necessary and sufficient conditions are shown for these relaxations to be tight. In particular, we prove they are tight under the cop-SOS convexity assumption. Compared to the traditional dense ones, the sparse Moment-SOS relaxations are more computationally efficient. Numerical experiments are given to show the efficiency.

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arXiv2026

Partial Lagrange Multiplier Expressions and Disjunctive Decompositions for Bilevel Optimization

This paper studies bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables. We show that such bilevel program can be reformulated as a disjunctive program by using Karush-Kuhn-Tucker (KKT) conditions with a sparse type of Lagrange multipliers. This kind of Lagrange multipliers can be conveniently represented by polynomials, for which we call partial Lagrange multiplier expressions (PLMEs). By doing this, each branch problem of the disjunctive program can be solved efficiently by polynomial optimization techniques. Solving each branch problem either returns infeasibility or gives a candidate local or global optimizer for the original bilevel optimization. We give necessary and sufficient conditions for these candidates to be global optimizers, and sufficient conditions for the local optimality. Numerical experiments are also presented to show the efficiency of the method.

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arXiv2026

Log-Polynomial Optimization

We study an optimization problem in which the objective is given as a sum of logarithmic-polynomial functions. This formulation is motivated by statistical estimation principles such as maximum likelihood estimation, and by loss functions including cross-entropy and Kullback-Leibler divergence. We propose a hierarchy of moment relaxations based on the truncated $K$-moment problems to solve log-polynomial optimization. We provide sufficient conditions for the hierarchy to be tight and introduce a numerical method to extract the global optimizers when the tightness is achieved. In addition, we modify relaxations with optimality conditions to better fit log-polynomial optimization with convenient Lagrange multipliers expressions. Various applications and numerical experiments are presented to show the efficiency of our method.

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arXiv2025

A global approach for generalized semi-infinte programs with polyhedral parameter sets

This paper studies generalized semi-infinite programs (GSIPs) defined with polyhedral parameter sets. Assume these GSIPs are given by polynomials. We propose a new approach to solve them as a disjunctive program. This approach is based on the Karush-Kuhn-Tucker (KKT) conditions of the robust constraint and a technique called partial Lagrange multiplier expressions. We summarize a semidefinite algorithm and study its convergence properties. Numerical experiments are given to show the efficiency of our method. In addition, we checked its performance in gemstone cutting and robust control applications.

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arXiv2025

Distributionally Robust Optimization with Polynomial Robust Constraints

This paper studies distributionally robust optimization (DRO) with polynomial robust constraints. We give a Moment-SOS relaxation approach to solve the DRO. This reduces to solving linear conic optimization with semidefinite constraints. When the DRO problem is SOS-convex, we show that it is equivalent to the linear conic relaxation and it can be solved by the Moment-SOS algorithm. For nonconvex cases, we also give concrete conditions such that the DRO can be solved globally. Numerical experiments are given to show the efficiency of the method.

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arXiv2024

Robust approximation of chance constrained optimization with polynomial perturbation

This paper proposes a robust approximation method for solving chance constrained optimization (CCO) of polynomials. Assume the CCO is defined with an individual chance constraint that is affine in the decision variables. We construct a robust approximation by replacing the chance constraint with a robust constraint over an uncertainty set. When the objective function is linear or SOS-convex, the robust approximation can be equivalently transformed into linear conic optimization. Semidefinite relaxation algorithms are proposed to solve these linear conic transformations globally and their convergent properties are studied. We also introduce a heuristic method to find efficient uncertainty sets such that optimizers of the robust approximation are feasible to the original problem. Numerical experiments are given to show the efficiency of our method.

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arXiv2024

Towards Global Solutions for Nonconvex Two-Stage Stochastic Programs: A Polynomial Lower Approximation Approach

This paper tackles the challenging problem of finding global optimal solutions for two-stage stochastic programs with continuous decision variables and nonconvex recourse functions. We introduce a two-phase approach. The first phase involves the construction of a polynomial lower bound for the recourse function through a linear optimization problem over a nonnegative polynomial cone. Given the complex structure of this cone, we employ semidefinite relaxations with quadratic modules to facilitate our computations. In the second phase, we solve a surrogate first-stage problem by substituting the original recourse function with the polynomial lower approximation obtained in the first phase. Our method is particularly advantageous for two reasons: it not only generates global lower bounds for the nonconvex stochastic program, aiding in the certificate of global optimality for prospective solutions like stationary solutions computed from other methods, but it also yields an explicit polynomial approximation for the recourse function through the solution of a linear conic optimization problem, where the number of variables is independent of the support of the underlying random vector. Therefore, our approach is particularly suitable for the case where the random vector follows a continuous distribution or when dealing with a large number of scenarios. Numerical experiments are conducted to demonstrate the effectiveness of our proposed approach.

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arXiv2023

A New Working Set Method for Nonlinear Inequality Constrained Minimization

The main purpose of this project is to develop a new active set method (ASM) called a working set method (WSM) for solving a general nonlinear inequality constrained minimization problem in a Hilber space. Mathematical analysis is carried out to validate the method and to show its merits over other ASMs. Since the method is quite general and new, some results on its implementation and application are presented. Numerical examples on some benchmark problems are carried out to test the method.

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arXiv2023

Global optimization for the portfolio selection model with high-order moments

In this paper, we study the global optimality of polynomial portfolio optimization (PPO). The PPO is a kind of portfolio selection model with high-order moments and flexible risk preference parameters. We introduce a perturbation sample average approximation method, which can give a robust approximation of the PPO in form of linear conic optimization. The approximated problem can be solved globally with Moment-SOS relaxations. We summarize a semidefinite algorithm, which can be used to find reliable approximations of the optimal value and optimizer set of the PPO. Numerical examples are given to show the efficiency of the algorithm.

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arXiv2023

Rational Generalized Nash Equilibrium Problems

This paper studies generalized Nash equilibrium problems that are given by rational functions. The optimization problems are not assumed to be convex. Rational expressions for Lagrange multipliers and feasible extensions of KKT points are introduced to compute a generalized Nash equilibrium (GNE). We give a hierarchy of rational optimization problems to solve rational generalized Nash equilibrium problems. The existence and computation of feasible extensions are studied. The Moment-SOS relaxations are applied to solve the rational optimization problems. Under some general assumptions, we show that the proposed hierarchy can compute a GNE if it exists or detect its nonexistence. Numerical experiments are given to show the efficiency of the proposed method.

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arXiv2022

Distributionally Robust Optimization with Moment Ambiguity Sets

This paper studies distributionally robust optimization (DRO) when the ambiguity set is given by moments for the distributions. The objective and constraints are given by polynomials in decision variables. We reformulate the DRO with equivalent moment conic constraints. Under some general assumptions, we prove the DRO is equivalent to a linear optimization problem with moment and psd polynomial cones. A Moment-SOS relaxation method is proposed to solve it. Its asymptotic and finite convergence are shown under certain assumptions. Numerical examples are presented to show how to solve DRO problems.

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arXiv2022

Dehomogenization for Completely Positive Tensors

A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the dehomogenization approach.

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arXiv2022

Loss Functions for Finite Sets

This paper studies loss functions for finite sets. For a given finite set $S$, we give sum-of-square type loss functions of minimum degree. When $S$ is the vertex set of a standard simplex, we show such loss functions have no spurious minimizers (i.e., every local minimizer is a global one). Up to transformations, we give similar loss functions without spurious minimizers for general finite sets. When $S$ is approximately given by a sample set $T$, we show how to get loss functions by solving a quadratic optimization problem. Numerical experiments and applications are given to show the efficiency of these loss functions.

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arXiv2021

A Lagrange Multiplier Expression Method for Bilevel Polynomial Optimization

This paper studies bilevel polynomial optimization problems. To solve them, we give a method based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment-SOS relaxations are used to solve the polynomial optimizattion relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimization problems. Numerical experiments are presented to show the efficiency of the method.

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arXiv2019

Stochastic Polynomial Optimization

This paper studies stochastic optimization problems with polynomials. We propose an optimization model with sample averages and perturbations. The Lasserre type Moment-SOS relaxations are used to solve the sample average optimization. Properties of the optimization and its relaxations are studied. Numerical experiments are presented.

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