Global Minimization via Piecewise-Linear Underestimation

Given a function on Rⁿ with many multiple local minima we approximate it from below, via concave minimization, with a piecewise-linear convex function by using sample points from the given function. The piecewise-linear function is then minimized using a single linear program to obtain an approximation to the global minimum of the original function. Successive shrinking of the original search region to which this procedure is applied leads to fairly accurate estimates, within 0.57%, of the global minima of synthetic nonconvex piecewise-quadratic functions for which the global minima are known exactly.     

Globally Convergent Variable Metric Method for Nonconvex Nondifferentiable Unconstrained Minimization

A special variable metric method is given for finding the stationary points of locally Lipschitz continuous functions which are not necessarily convex or differentiable. Time consuming quadratic programming subproblems do not need to be solved. Global convergence of the method is established. Some encouraging numerical experience is reported.     

脉冲噪声下图像去模糊非光滑非凸最小化问题的近端线性化最小化算法

去除脉冲噪声是图像复原中的重要任务之一.我们提出一类非光滑非凸模型来恢复模糊和脉冲噪声污染的图像,该模型具有灵活的先验信息引入机制,如盒子约束或低秩等.为了求解所提非凸问题,我们采用近端线性化最小化算法.对于算法中的子问题,我们运用交替方向乘子法.在目标函数满足Kurdyka-Lojasiewicz性质的假设下,我们证明所提算法的全局收敛性.数值实验表明,在主观和客观质量评价方面,我们的方法优于$\ell_{1}$TV和非凸TV模型.     

On the Dynamics of a Differential Inclusion Built upon a Nonconvex Constrained Minimization Problem

In this paper, we study the dynamics of a differential inclusion built upon a nonsmooth, not necessarily convex, constrained minimization problem in finite-dimensional spaces. In particular, we are interested in the investigation of the asymptotic behavior of the trajectories of the dynamical system represented by the differential inclusion. Under suitable assumptions on the constraint set and the two involved functions (one defining the constraint set, the other representing the functional to be minimized), it is proved that all the trajectories converge to the set of the constrained critical points. We present also a large class of constraint sets satisfying our assumptions. As a simple consequence, in the case of a smooth convex minimization problem, we have that any trajectory converges to the set of minimizers.Research partially supported by the research project CNR-GNAMPA Mathematical Methods for Control Theory and supported in part by the European Communitys Human Potential Program under Contract HPRN-CT-2002-00281, Evolution Equations.     

Convex Kernel Underestimation of Functions with Multiple Local Minima

A function on Rⁿ with multiple local minima is approximated from below, via linear programming, by a linear combination of convex kernel functions using sample points from the given function. The resulting convex kernel underestimator is then minimized, using either a linear equation solver for a linear-quadratic kernel or by a Newton method for a Gaussian kernel, to obtain an approximation to a global minimum of the original function. Successive shrinking of the original search region to which this procedure is applied leads to fairly accurate estimates, within 0.0001% for a Gaussian kernel function, relative to global minima of synthetic nonconvex piecewise-quadratic functions for which the global minima are known exactly. Gaussian kernel underestimation improves by a factor of ten the relative error obtained using a piecewise-linear underestimator (O.L. Mangasarian, J.B. Rosen, and M.E. Thompson, Journal of Global Optimization, Volume 32, Number 1, Pages 1–9, 2005), while cutting computational time by an average factor of over 28.     

A parametric successive underestimation method for convex multiplicative programming problems

This paper addresses itself to the algorithm for minimizing the product of two nonnegative convex functions over a convex set. It is shown that the global minimum of this nonconvex problem can be obtained by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in a higher dimensional space and to apply a branch-and-bound algorithm using an underestimating function. Computational results indicate that our algorithm is efficient when the objective function is the product of a linear and a quadratic functions and the constraints are linear. An extension of our algorithm for minimizing the sum of a convex function and a product of two convex functions is also discussed.     

Stability Index Method for Global Minimization

The Stability Index Method (SIM) combines stochastic and deterministic algorithms to find global minima of multidimensional functions. The functions may be nonsmooth and may have multiple local minima. The method examines the change of the diameters of the minimizing sets for its stopping criterion. At first, the algorithm uses the uniform random distribution in the admissible set. Then normal random distributions of decreasing variation are used to focus on probable global minimizers. To test the method, it is applied to seven standard test functions of several variables. The computational results show that the SIM is efficient, reliable and robust.The authors thank the referees for valuable suggestions.     

半定矩阵秩极小的非凸精确松弛

本文主要研究半定矩阵秩极小问题(P)的非凸精确松弛及其性质.首先,为求解问题(P),我们引入其Schatten p-范数(0＜p＜1)松弛,记为(Sp).其次,通过定义半定限制等距常数和半定限制正交常数,我们给出了问题(P)有唯—解的充分条件.最后,利用半定限制等距性质,我们给出了问题(P)和(Sp)有相同唯一解的充分条件.特别地,对任意0＜p＜1,我们还得到—个一致的精确恢复条件.     

Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization

The subject of this article is a class of global optimization problems, in which the variables can be divided into two groups such that, in each group, the functions involved have the same structure (e.g. linear, convex or concave, etc.). Based on the decomposition idea of Benders (Ref. 1), a corresponding master problem is defined on the space of one of the two groups of variables. The objective function of this master problem is in fact the optimal value function of a nonlinear parametric optimization problem. To solve the resulting master problem, a branch-and-bound scheme is proposed, in which the estimation of the lower bounds is performed by applying the well-known weak duality theorem in Lagrange duality. The results of this article concentrate on two subjects: investigating the convergence of the general algorithm and solving dual problems of some special classes of nonconvex optimization problems. Based on results in sensitivity and stability theory and in parametric optimization, conditions for the convergence are established by investigating the so-called dual properness property and the upper semicontinuity of the objective function of the master problem. The general algorithm is then discussed in detail for some nonconvex problems including concave minimization problems with a special structure, general quadratic problems, optimization problems on the efficient set, and linear multiplicative programming problems.     

An Interior-Point Algorithm for Nonconvex Nonlinear Programming

The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.     

Robust Solution of Nonconvex Global Optimization Problems

The concept of ɛ-approximate optimal solution as widely used in nonconvex global optimization is not quite adequate, because such a point may correspond to an objective function value far from the true optimal value, while being infeasible. We introduce a concept of essential ɛ-optimal solution, which gives a more appropriate approximate optimal solution, while being stable under small perturbations of the constraints. A general method for finding an essential ɛ-optimal solution in finitely many steps is proposed which can be applied to d.c. programming and monotonic optimization.     

Two-Phase Image Segmentation by Nonconvex Nonsmooth Models with Convergent Alternating Minimization Algorithms

Two-phase image segmentation is a fundamental task to partition an image into foreground and background. In this paper, two types of nonconvex and nonsmooth regularization models are proposed for basic two-phase segmentation. They extend the convex regularization on the characteristic function on the image domain to the nonconvex case, which are able to better obtain piecewise constant regions with neat boundaries. By analyzing the proposed non-Lipschitz model, we combine the proximal alternating minimization framework with support shrinkage and linearization strategies to design our algorithm. This leads to two alternating strongly convex subproblems which can be easily solved. Similarly, we present an algorithm without support shrinkage operation for the nonconvex Lipschitz case. Using the Kurdyka-Łojasiewicz property of the objective function, we prove that the limit point of the generated sequence is a critical point of the original nonconvex nonsmooth problem. Numerical experiments and comparisons illustrate the effectiveness of our method in two-phase image segmentation.     

解非凸规划问题动边界组合同伦方法

本文给出了一个新的求解非凸规划问题的同伦方法,称为动边界同伦方程,并在较弱的条件下,证明了同伦路径的存在性和大范围收敛性．与已有的拟法锥条件、伪锥条件下的修正组合同伦方法相比,同伦构造更容易,并且不要求初始点是可行集的内点,因此动边界组合同伦方法比修正组合同伦方法及弱法锥条件下的组合同伦内点法和凝聚约束同伦方法更便于应用．     

Iterative Schemes for Nonconvex Variational Inequalities

In this paper, we suggest and analyze some iterative methods for solving nonconvex variational inequalities using the auxiliary principle technique, the convergence of which requires either only pseudomonotonicity or partially relaxed strong monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving general variational inequalities involving convex sets.     

Lagrangian Duality for Minimization of Nonconvex Multifunctions

Two alternative type theorems for nearly convexlike or * quasiconvex multifunctions are presented. They are used to derive Lagrangian conditions and duality results for vector optimization problems when the objectives and the constraints are nearly convexlike or * quasiconvex multifunctions.     

A Hybrid Genetic Algorithm for Nonconvex Function Minimization

In this paper, we consider the problem of minimizing a function in severalvariables which could be multimodal and may possess discontinuities. A newalgorithm for the problem based on the genetic technique is developed. Thealgorithm is hybrid in nature in the sense that it utilizes the genetictechnique to generate search directions, which are used in an optimizationscheme and is thus different from any other methods in the literature.The algorithm has been tested on the Rosenbrock valley functions in 2 and 4dimensions, and multimodal functions in 2 and 4 dimensions, which are of ahigh degree of difficulty. The results are compared with the Adaptive RandomSearch, and Simulated Annealing algorithms. The performance of the algorithmis also compared to recent global algorithms in terms of the number offunctional evaluations needed to obtain a global minimum and results show thatthe proposed algorithm is better than these algorithms on a set of standardtest problems. It seems that the proposed algorithm is efficient and robust.     

同伦方法求解非凸区域Brouwer不动点问题

本文构造了一个新的求解非凸区域上不动点问题的内点同伦算法,并在弱法锥(见定义2．1(2))和适当的条件下,证明了算法的全局收敛性．本文所给的条件比外法锥条件更加一般．     

Optimality Conditions for Nonconvex Constrained Optimization Problems

Abstract

We present optimality conditions for a class of nonsmooth and nonconvex constrained optimization problems. To achieve this aim, various well-known constraint qualifications are extended based on the concept of tangential subdifferential and the relations between them are investigated. Moreover, local and global necessary and sufficient optimality conditions are derived in the absence of convexity of the feasible set. In addition to the theoretical results, several examples are provided to illustrate the advantage of our outcomes.     

非凸半定规划的增广Lagrangian的微分的计算

迄今为止，还未见出版过有关求解非凸半定规划的算法，但在最近，Chen，et.al(2000)和Sun＆Sun(1999)关于非凸半定规划(SDP)的增广Lagrangian的研究是非常有用的，在本文中，我们证明非凸半定规划的增广Lagrangian是可微的，并且给出它的可微表达式．