On methods for solving nonlinear semidefinite optimization problems
Jie Sun
Numerical Algebra, Control & Optimization 2011, 1(1): 1-14 doi: 10.3934/naco.2011.1.1
The nonlinear semidefinite optimization problem arises from applications in system control, structural design, financial management, and other fields. However, much work is yet to be done to effectively solve this problem. We introduce some new theoretical and algorithmic development in this field. In particular, we discuss first and second-order algorithms that appear to be promising, which include the alternating direction method, the augmented Lagrangian method, and the smoothing Newton method. Convergence theorems are presented and preliminary numerical results are reported.
keywords: Alternating direction method augmented Lagrangian method semismooth Newton method.
A smoothing Newton algorithm for mathematical programs with complementarity constraints
Zheng-Hai Huang Jie Sun
Journal of Industrial & Management Optimization 2005, 1(2): 153-170 doi: 10.3934/jimo.2005.1.153
We propose a smoothing Newton algorithm for solving mathematical programs with complementarity constraints (MPCCs). Under some reasonable conditions, the proposed algorithm is shown to be globally convergent and to generate a $B$-stationary point of the MPCC. Preliminary numerical results on some MacMPEC problems are reported.
keywords: global convergence. mathematical program with complementarity constraints B-stationary point smoothing algorithm
Designing the distribution network for an integrated supply chain
Jia Shu Jie Sun
Journal of Industrial & Management Optimization 2006, 2(3): 339-349 doi: 10.3934/jimo.2006.2.339
We consider an integrated distribution network design problem in which all the retailers face uncertain demand. The risk-pooling benefit is achieved by allowing some of the retailers to operate as distribution centers (DCs) with commitment in service level. The target is to minimize the expected total cost resulted from the DC location, transportation, and inventory. We formulate it as a two-stage nonlinear discrete stochastic optimization problem. The first stage decides which retailers to be selected as DCs and the second stage deals with the costs of DC-retailer assignment, transportation, and inventory. In the literature, the similar models require the demands of all retailers in each scenario to have their variances identically proportional to their means. In this paper, we remove this restriction. We reformulate the problem as a set-covering model and solve it by a column generation approach. With a variable fixing technique, we are able to efficiently solve problems of moderate-size (up to one hundred retailers and nine scenarios).
keywords: supply chain design. set covering Column generation
A class of two-stage distributionally robust games
Bin Li Jie Sun Honglei Xu Min Zhang
Journal of Industrial & Management Optimization 2019, 15(1): 387-400 doi: 10.3934/jimo.2018048

An $ N $-person noncooperative game under uncertainty is analyzed, in which each player solves a two-stage distributionally robust optimization problem that depends on a random vector as well as on other players' decisions. Particularly, a special case is considered, where the players' optimization problems are linear at both stages, and it is shown that the Nash equilibrium of this game can be obtained by solving a conic linear variational inequality problem.

keywords: Conic optimization duality game with uncertainty
Analysis of optimal boundary control for a three-dimensional reaction-diffusion system
Wanli Yang Jie Sun Su Zhang
Numerical Algebra, Control & Optimization 2017, 7(3): 325-344 doi: 10.3934/naco.2017021

This paper is concerned with optimal boundary control of a three dimensional reaction-diffusion system in a more general form than what has been presented in the literature. The state equations are analyzed and the optimal control problem is investigated. Necessary and sufficient optimality conditions are derived. The model is widely applicable due to its generality. Some examples in applications are discussed.

keywords: Optimal control reaction-diffusion system optimality conditions
Inertial accelerated algorithms for solving a split feasibility problem
Yazheng Dang Jie Sun Honglei Xu
Journal of Industrial & Management Optimization 2017, 13(3): 1383-1394 doi: 10.3934/jimo.2016078

Inspired by the inertial proximal algorithms for finding a zero of a maximal monotone operator, in this paper, we propose two inertial accelerated algorithms to solve the split feasibility problem. One is an inertial relaxed-CQ algorithm constructed by applying inertial technique to a relaxed-CQ algorithm, the other is a modified inertial relaxed-CQ algorithm which combines the KM method with the inertial relaxed-CQ algorithm. We prove their asymptotical convergence under some suitable conditions. Numerical results are reported to show the effectiveness of the proposed algorithms.

keywords: Asymptotical convergence inertial technique nonsmooth analysis split feasibility problem subdifferential
Double projection algorithms for solving the split feasibility problems
Ya-zheng Dang Jie Sun Su Zhang
Journal of Industrial & Management Optimization 2017, 13(5): 1-12 doi: 10.3934/jimo.2018135

We propose two new double projection algorithms for solving the split feasibility problem (SFP). Different from the extragradient projection algorithms, the proposed algorithms do not require fixed stepsize and do not employ the same projection region at different projection steps. We adopt flexible rules for selecting the stepsize and the projection region. The proposed algorithms are shown to be convergent under certain assumptions. Numerical experiments show that the proposed methods appear to be more efficient than the relaxed- CQ algorithm.

keywords: Armijo-type line search convergence analysis double projection algorithm optimization split feasibility problem

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