Feasible Semismooth Newton Method for a Class of Stochastic Linear Complementarity Problems

We consider a class of stochastic linear complementarity problems (SLCPs) with finitely many realizations. In this paper we reformulate this class of SLCPs as a constrained minimization (CM) problem. Then, we present a feasible semismooth Newton method to solve this CM problem. Preliminary numerical results show that this CM reformulation may yield a solution with high safety for SLCPs.     

New reformulation and feasible semismooth Newton method for a class of stochastic linear complementarity problems

A class of stochastic linear complementarity problems (SLCPs) with finitely many realizations is considered. We first formulate the problem as a new constrained minimization problem. Then, we propose a feasible semismooth Newton method which yields a stationary point of the constrained minimization problem. We study the condition for the level set of the objective function to be bounded. As a result, the condition for the solution set of the constrained minimization problem is obtained. The global and quadratic convergence of the proposed method is proved under certain assumptions. Preliminary numerical results show that this method yields a reasonable solution with high safety and within a small number of iterations.     

随机线性互补问题的无约束优化再定式

针对随机线性互补问题,提出等价的无约束优化再定式模型,即由D-间隙函数定义的确定性的无约束期望残差极小化问题.通过拟Monte Carlo方法,将样本进行了推广,得到了相关的离散近似问题.在适当的条件下,提出了最优解存在的充分条件,以及探究了离散近似问题的最优解及稳定点的收敛性.另外,在针对一类带有常系数矩阵的随机互补线性问题,研究了解存在的充要条件.     

NONDESCENT SUBGRADIENT METHOD FOR NONSMOOTH CONSTRAINED MINIMIZATION

A kind of nondecreasing subgradient algorithm with appropriate stopping rule has been proposed for nonsmooth constrained minimization problem. The dual theory is invoked in dealing with the stopping rule and general global minimiizing algorithm is employed as a subroutine of the algorithm. The method is expected to tackle a large class of nonsmooth constrained minimization problem.     

Hidden Convex Minimization

A class of nonconvex minimization problems can be classified as hidden convex minimization problems. A nonconvex minimization problem is called a hidden convex minimization problem if there exists an equivalent transformation such that the equivalent transformation of it is a convex minimization problem. Sufficient conditions that are independent of transformations are derived in this paper for identifying such a class of seemingly nonconvex minimization problems that are equivalent to convex minimization problems. Thus, a global optimality can be achieved for this class of hidden convex optimization problems by using local search methods. The results presented in this paper extend the reach of convex minimization by identifying its equivalent with a nonconvex representation.     

A smoothing Levenberg-Marquardt algorithm for solving a class of stochastic linear complementarity problem

In this paper, it is considered for a class of stochastic linear complementarity problems (SLCPs) with finitely many elements. A smoothing Levenberg-Marquardt algorithm is proposed for solving the SLCP. Under suitable conditions, the global convergence and local quadratic convergence of the proposed algorithm is given. Some numerical results are reported in this paper, which confirms the good theoretical properties of the proposed algorithm.     

A FEASIBLE SEMISMOOTH GAUSS-NEWTON METHOD FOR SOLVING A CLASS OF SLCPS

In this paper, we consider a class of the stochastic linear complementarity problems (SLCPs) with finitely many elements. A feasible semismooth damped Gauss-Newton algorithm for the SLCP is proposed. The global and local quadratic convergence of the proposed algorithm are obtained under suitable conditions. Some numerical results are reported in this paper, which confirm the good theoretical properties of the proposed algorithm.     

Parallel Nesterov’s method for large-scale minimization of partially separable functions

A parallel Nesterov algorithm, for solving unconstrained minimization of large scale partially separable convex functions, is presented. The problem is first transformed into a linearly constrained minimization of a separable function. A fast projected gradient (Nesterov) method is then applied to obtain a decomposition method with \(O(1/k^2)\) rate of convergence (where k is the iteration number). Preliminary numerical experiments show the efficiency of the proposed approach.     

A smoothing Newton method for solving a class of stochastic linear complementarity problems

In this paper, we consider a class of stochastic linear complementarity problems (SLCPs) with finitely many elements. We present a smoothing Newton algorithm for solving the SLCP. Under suitable conditions, we obtain the global convergence and locally quadratic convergence of the proposed algorithm. Some numerical results are reported in this paper, which confirm the good theoretical properties of the proposed algorithm.     

An efficient solution method for rank two quasiconcave minimization problems

The technique of dimension reduction earlier developed by the first author is applied to the class of nonconvex minimization problems having the so called rank two property. This class includes in particular the problem of minimizing the product of two affine functions over a polytope. An efficient method for solving this class of problems is presented. Also some results of computational experiments with this method are discussed     

带新NCP函数的Lagrangian乘子方法

在经营管理、工程设计、科学研究、军事指挥等方面普遍存在着最优化问题,而实际问题中出现的绝大多数问题都被归纳为非线性规划问题之中。作为带等式、不等式约束的复杂事例,最优化问题的求解向来较为繁琐、困难。适当条件下,非线性互补函数(NCP)可以与约束优化问题相结合,其中NCP函数的无约束极小解对应原约束问题的解及其乘子。本文提出了一类新的NCP函数用于解决等式和不等式约束非线性规划问题,结合新的NCP函数构造了增广Lagrangian函数。在适当假设条件下,证明了增广Lagrangian函数与原问题的解之间的一一对应关系。同时构造了相应算法,并证明了该算法的收敛性和有效性。     

Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities

This paper considers a class of vector variational inequalities. First, we present an equivalent formulation, which is a scalar variational inequality, for the deterministic vector variational inequality. Then we concentrate on the stochastic circumstance. By noting that the stochastic vector variational inequality may not have a solution feasible for all realizations of the random variable in general, for tractability, we employ the expected residual minimization approach, which aims at minimizing the expected residual of the so-called regularized gap function. We investigate the properties of the expected residual minimization problem, and furthermore, we propose a sample average approximation method for solving the expected residual minimization problem. Comprehensive convergence analysis for the approximation approach is established as well.     

A quadratically constrained minimization problem arising from PDE of Monge–Ampère type

This note develops theory and a solution technique for a quadratically constrained eigenvalue minimization problem. This class of problems arises in the numerical solution of fully-nonlinear boundary value problems of Monge–Ampère type. Though it is most important in the three dimensional case, the solution method is directly applicable to systems of arbitrary dimension. The focus here is on solving the minimization subproblem which is part of a method to numerically solve a Monge–Ampère type equation. These subproblems must be evaluated many times in this numerical solution technique and thus efficiency is of utmost importance. A novelty of this minimization algorithm is that it is finite, of complexity O(n³)\mathcal{O}(n^3), with the exception of solving a very simple rational function of one variable. This function is essentially the same for any dimension. This result is quite surprising given the nature of the constrained minimization problem.     

Minimization of equilibrium problems,variational inequality problems and fixed point problems

In this paper, we devote to find the solution of the following quadratic minimization problem

$\min_{x\in \Omega}\|x\|^2,$

where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping and the solution set of some variational inequality. In order to solve the above minimization problem, we first construct an implicit algorithm by using the projection method. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. Finally, we prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem.

     

A New Approximation of the Matrix Rank Function and Its Application to Matrix Rank Minimization

The matrix rank minimization problem is widely applied in many fields such as control, signal processing and system identification. However, the problem is NP-hard in general and is computationally hard to directly solve in practice. In this paper, we provide a new approximation function of the matrix rank function, and the corresponding approximation problems can be used to approximate the matrix rank minimization problem within any level of accuracy. Furthermore, the successive projected gradient method, which is designed based on the monotonicity and the Fréchet derivative of these new approximation function, can be used to solve the matrix rank minimization this problem by using the projected gradient method to find the stationary points of a series of approximation problems. Finally, the convergence analysis and the preliminary numerical results are given.     

An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems

We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems.     

On merit functions for <Emphasis Type="Italic">p</Emphasis>-order cone complementarity problem

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.     

A convexification method for a class of global optimization problems with applications to reliability optimization

A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimization problems using the proposed convexification schemes. An outer approximation method can then be used to find the global solution of the transformed problem. Applications to mixed-integer nonlinear programming problems arising in reliability optimization of complex systems are discussed and satisfactory numerical results are presented.     

Application of the hypodifferential descent method to the problem of constructing an optimal control

In this paper the problem of optimal control of a nonlinear ODE system with given boundary conditions and the integral restriction on control is considered. With the help of the theory of exact penalty functions the original problem is reduced to the problem of unconstrained minimization of a nonsmooth functional. The necessary minimum conditions in terms of hypodifferentials are found. A class of problems for which these conditions are also sufficient is distinguished. On the basis of these conditions the hypodifferential descent method is applied to the considered problem. Under some additional assumptions the hypodifferential descent method converges in a certain sense.     

An algorithm for the estimation of a regression function by continuous piecewise linear functions

The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L ₂ risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.