arXiv subjects

Liguo Jiao

Publications and source records attributed to Liguo Jiao.

16 recordsLinked to original sources

arXiv2026

Second-Order Optimality Conditions for Sparse Differentiable Optimization Problems via Limiting Second-Order Subdifferentials

In this paper, we investigate a class of sparse optimization problems in which both the objective and constraint functions are Fr\'echet differentiable and possess locally Lipschitz continuous gradient mappings. More precisely, by utilizing the limiting (Mordukhovich) second-order subdifferential of the associated Lagrangian function, we establish new second-order necessary and sufficient optimality conditions for local optimal solutions. The obtained results are derived under mild assumptions and extend several existing results in the literature. In addition, we apply our theoretical developments to sparse multiobjective optimization problems and derive second-order sufficient optimality conditions for efficient solutions. Several examples are also presented to demonstrate the applicability and effectiveness of the proposed results.

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arXiv2025

The charnes--cooper transformation and fractional programs with convex polynomials

This paper proposes a scheme based on the Charnes--Cooper transformation for solving a class of fractional programs with convex polynomial data. We employ the high-degree perturbation method to construct relaxations based on sums of squares for the fractional programs under consideration. Under suitable conditions, we establish results on solution existence, strong duality, asymptotic convergence, and solution extraction. As a special case, fractional programs with SOS-convex polynomial data are also studied.

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arXiv2025

First-order SDSOS-convex semi-algebraic optimization and exact SOCP relaxations

In this paper, we define a new type of nonsmooth convex function, called {\em first-order SDSOS-convex semi-algebraic function}, which is an extension of the previously proposed first-order SDSOS-convex polynomials (Chuong et al. in J Global Optim 75:885--919, 2019). This class of nonsmooth convex functions contains many well-known functions, such as the Euclidean norm, the $\ell_1$-norm commonly used in compressed sensing and sparse optimization, and the least squares function frequently employed in machine learning and regression analysis. We show that, under suitable assumptions, the optimal value and optimal solutions of first-order SDSOS-convex semi-algebraic programs can be found by exactly solving an associated second-order cone programming problem. Finally, an application to robust optimization with first-order SDSOS-convex polynomials is discussed.

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arXiv2025

Exact penalty functions in optimization with unbounded constraint sets

This paper identifies necessary and sufficient conditions for the exactness of penalty functions in optimization problems whose constraint sets are not necessarily bounded. The case where the data of problems is locally Lipschitz, semi-algebraic or non-degenerate polynomials is studied in detail. The conditions are given in terms of properties of the objective and residual functions of the problems in question. The obtained results generalize and improve some known results in the literature on exact penalty functions.

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arXiv2025

SPLD polynomial optimization and bounded degree SOS hierarchies

In this paper, we introduce a new class of structured polynomials, called separable plus lower degree (SPLD) polynomials. The formal definition of an SPLD polynomial, which extends the concept of SPQ polynomials (Ahmadi et al. in Math Oper Res 48:1316--1343, 2023), is provided. A type of bounded degree SOS hierarchy, referred to as BSOS-SPLD, is proposed to efficiently solve optimization problems involving SPLD polynomials. Numerical experiments on several benchmark problems indicate that the proposed method yields better performance than the standard bounded degree SOS hierarchy (Lasserre et al. in EURO J Comput Optim 5:87--117, 2017). An exact SOS relaxation for a class of convex SPLD polynomial optimization problems is proposed. Finally, we present an application of SPLD polynomials to convex polynomial regression problems arising in statistics.

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arXiv2024

Approximate optimality conditions and sensitivity analysis in nearly convex optimization

In this paper, approximate optimality conditions and sensitivity analysis in nearly convex optimization are discussed. More precisely, as in the spirit of convex analysis, we introduce the concept of $\varepsilon$-subdifferential for nearly convex functions. Then, we examine some significant properties and rules for the $\varepsilon$-subdifferential. These rules are applied to study optimality conditions as well as sensitivity analysis for parametric nearly convex optimization problems, which are two important topics in optimization theory.

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arXiv2024

A parameter-free approach for solving SOS-convex semi-algebraic fractional programs

In this paper, we study a class of nonsmooth fractional programs {\rm (FP, for short)} with SOS-convex semi-algebraic functions. Under suitable assumptions, we derive a strong duality result between the problem (FP) and its semidefinite programming (SDP) relaxations. Remarkably, we extract an optimal solution of the problem (FP) by solving one and only one associated SDP problem. Numerical examples are also given.

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arXiv2022

Utopia point method based robust vector polynomial optimization scheme

In this paper, we focus on a class of robust vector polynomial optimization problems (RVPOP in short) without any convex assumptions. By combining/improving the utopia point method (a nonlinear scalarization) for vector optimization and "joint+marginal" relaxation method for polynomial optimization, we solve the RVPOP successfully. Both theoratical and computational aspects are considered.

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arXiv2022

A new scheme for approximating the weakly efficient solution set of vector rational optimization problems

In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More precisely, we present a procedure to obtain a sequence of explicit approximations of the weakly efficient solution set of the problem in question. Each approximation is the intersection of the sublevel set of a single polynomial and the feasible set. To this end, we make use of the achievement function associated with the considered problem and construct polynomial approximations of it over the feasible set from above. Remarkably, the construction can be converted to semidefinite programming problems. Several nontrivial examples are designed to illustrate the proposed new scheme.

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arXiv2022

Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints

In this paper, we are interested in the existence of Pareto solutions to vector polynomial optimization problems over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called {\it tangency varieties}; then we establish connections of the Palais--Smale condition, Cerami condition, {\it M}-tameness, and properness related to the considered problem, in which the condition of Mangasarian--Fromovitz constraint qualification at infinity plays an essential role in deriving these connections. According to the obtained connections, we provide some sufficient conditions for existence of Pareto solutions to the problem in consideration, and we also give some examples to illustrate our main findings.

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arXiv2021

On Solving a Class of Fractional Semi-infinite Polynomial Programming Problems

In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex polynomial inequalities. To solve such a problem, we first reformulate it to a pair of primal and dual conic optimization problems, which reduce to semidefinite programming (SDP) problems if we can bring sum-of-squares structures into the conic constraints. To this end, we provide a characteristic cone constraint qualification for convex semi-infinite programming problems to guarantee strong duality and also the attainment of the solution in the dual problem, which is of its own interest. In this framework, we first present a hierarchy of SDP relaxations with asymptotic convergence for the FSIPP problem whose index set is defined by finitely many polynomial inequalities. Next, we study four cases of the FSIPP problems which can be reduced to either a single SDP problem or a finite sequence of SDP problems, where at least one minimizer can be extracted. Then, we apply this approach to the four corresponding multi-objective cases to find efficient solutions.

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arXiv2020

Multi-objective convex polynomial optimization and semidefinite programming relaxations

This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem (P$_z$) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem (P$_z$), we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite convergence results of the Lasserre-type hierarchy of semidefinite programming relaxations for problem (P$_z$) under suitable assumptions. Finally, we show that finding efficient solutions to problem (MP) can be achieved successfully by solving hierarchies of semidefinite programming relaxations and checking a flat truncation condition.

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arXiv2020

On continuous selections of polynomial functions

A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that there are only finitely many continuous selections of it and each one is semi-algebraic. Then, we establish some generic properties regarding the critical points, defined by the Clarke subdifferential, of these continuous selections. In particular, given a set of finitely many polynomials with generic coefficients, we show that the critical points of all continuous selections of it are finite and the critical values are all different, and we also derive the coercivity of those continuous selections which are bounded from below. We point out that some existing results about Łojasiewicz's inequality and error bounds for the maximum function of some finitely many polynomials are also valid for all the continuous selections of them.

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arXiv2020

On types of KKT points in polynomial optimization

Let $f$ be a real polynomial function with $n$ variables and $S$ be a basic closed semialgebraic set in $\Bbb{R}^n$. In this paper, we are interested in the problem of identifying the type (local minimizer, maximizer or not extremum point) of a given isolated KKT point $x^*$ of $f$ over $S.$ To this end, we investigate some properties of the tangency variety of $f$ on $S$ at $x^*,$ by which we introduce the definition of faithful radius of $f$ over $S$ at $x^*.$ Then, we show that the type of $x^*$ can be determined by the global extrema of $f$ over the intersection of $S$ and the Euclidean ball centered at $x^*$ with a faithful radius. Finally, we propose an algorithm involving algebraic computations to compute a faithful radius of $x^*$ and determine its type.

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arXiv2019

Polynomial mathematical programs with equilibrium constraints and semidefinite programming relaxations

This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a hierarchy of semidefinite programming (SDP) relaxations and prove the convergence result for the method. Numerical experiments are presented to show the efficiency of the proposed algorithm.

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arXiv2015

Convergence analysis of a proximal-type algorithm for DC programs with applications to variable selection

We consider a minimization problem of the form $P(\varphi, g, h):$ $$\min\left\{f(x):= \varphi(x) + g(x) - h(x) \colon x \in \mathbb{R}^n\right\},$$ where $\varphi$ is a differentiable function and $g,$ $h$ are convex functions, and introduce iterative methods to finding a critical point of $f$ when $f$ is differentiable. We show that the point computed by proximal point algorithm at each iteration can be used to determine a descent direction for the objective function at this point. This algorithm can be considered as a combination of proximal point algorithm together with a linesearch step that uses this descent direction. We also study convergence results of these algorithms and the inertial proximal methods proposed by Maing$\acute{e}$ and Moudafi (SIAM J. Optim. {\bf 19}(2008), 397--413) under the main assumption that the objective function satisfies the Kurdika--{\L}ojasiewicz property. The proposed algorithm is then applied to solve the variable selection problem in linear regression.

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