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An inexact Newton method for nonconvex equality constrained optimization

MATHEMATICAL programming 2010年第2期122卷 273-299页

作者： Byrd, Richard H. Curtis, Frank E. Nocedal, Jorge Northwestern Univ Dept Ind Engn & Management Sci Evanston IL 60208 USA Univ Colorado Dept Comp Sci Boulder CO 80309 USA Northwestern Univ Dept Elect Engn & Comp Sci Evanston IL 60208 USA

We present a matrix-free line search algorithm for large-scale equality constrained optimization that allows for inexact step computations. For strictly convex problems, the method reduces to the inexact sequential quadratic programming approach proposed by Byrd et al. [SIAM J. Optim. 19(1) 351-369, 2008]. For nonconvex problems, the methodology developed in this paper allows for the presence of negative curvature without requiring information about the inertia of the primal-dual iteration matrix. Negative curvature may arise from second-order information of the problem functions, but in fact exact second derivatives are not required in the approach. The complete algorithm is characterized by its emphasis on sufficient reductions in a model of an exact penalty function. We analyze the global behavior of the algorithm and present numerical results on a collection of test problems.

关键词： Large-scale optimization Constrained optimization nonconvex programming Inexact linear system solvers Krylov subspace methods

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Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in R²/Z²

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ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 2010年第3期27卷 837-856页

作者： Cardaliaguet, Pierre Univ Brest CNRS UMR 6205 Math Lab F-29285 Brest France

The paper investigates the long time average of the solutions of Hamilton-Jacobi equations with a noncoercive. nonconvex Hamiltonian in the torus R-2/Z(2). We give nonresonance conditions under which the long-time average converges to a constant. In the resonant case, we show that the limit still exists, although it is nonconstant in general. We compute the limit at points where it is not locally constant. (C) 2009 Elsevier Masson SAS. All rights reserved.

关键词： ERGODIC theory nonconvex programming HAMILTON-Jacobi equations TORUS (Geometry) EQUATIONS

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Solving a class of low rank d.c. programs via a branch and bound approach: A computational experience

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OPERATIONS RESEARCH LETTERS 2010年第5期38卷 354-357页

作者： Cambini, Riccardo Salvi, Francesca Univ Pisa Dept Stat & Appl Math Fac Econ I-56124 Pisa Italy

Various classes of d.c. programs have been studied in the recent literature due to their importance in applicative problems. In this paper we consider a branch and bound approach for solving a class of d.c. problems. Both stack policies and partitioning rules are analyzed, pointing out their performance effectiveness by means of the results of a computational experience. (C) 2010 Elsevier B.V. All rights reserved.

关键词： d.c. programmings nonconvex programming Branch and bound

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STRUCTURED TOTAL MAXIMUM LIKELIHOOD: AN ALTERNATIVE TO STRUCTURED TOTAL LEAST SQUARES

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2010年第5期31卷 2623-2649页

作者： Beck, Amir Eldar, Yonina C. Technion Israel Inst Technol Dept Ind Engn & Management IL-32000 Haifa Israel Technion Israel Inst Technol Dept Elect Engn IL-32000 Haifa Israel

Linear inverse problems with uncertain measurement matrices appear in many different applications. One of the standard techniques for solving such problems is the total least squares (TLS) method. Recently, an alternative approach has been suggested, based on maximizing an appropriate likelihood function assuming that the measurement matrix consists of random Gaussian variables. We refer to this technique as the total maximum likelihood (TML) method. Here we extend this strategy to the case in which the measurement matrix is structured so that the perturbations are not arbitrary but rather follow a fixed pattern. The resulting estimate is referred to as the structured TML (STML). As we show, the STML can be viewed as a regularized version of the structured TLS (STLS) approach in which the regularization consists of a logarithmic penalty. In contrast to the STLS solution, the STML always exists. Furthermore, its performance in practice tends to be superior to that of the STLS and competitive to other regularized solvers, as we illustrate via several examples. We also consider a few interesting special cases in which the STML can be computed efficiently either by reducing it into a one-dimensional problem regardless of the problem size or by a decomposition via a discrete Fourier transform.

关键词： total least squares maximum likelihood nonconvex programming linear inverse problem circulant structures

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On concave constraint functions and duality in predominantly black-and-white topology optimization

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2010年第33-36期199卷 2224-2234页

作者： Wood, Derren W. Groenwold, Albert A. Univ Stellenbosch Dept Mech Engn ZA-7600 Stellenbosch South Africa

We study the 'classical' discrete, solid-void or black-and-white topology optimization problem, in which minimum compliance is sought, subject to constraints on the available material resource. We assume that this problem is solved using methods that relax the discreteness requirements during intermediate steps, and that the associated programming problems are solved using sequential approximate optimization (SAO) algorithms based on duality. More specifically, we assume that the advantages of the well-known Falk dual are exploited. Such algorithms represent the state-of-the-art in (large-scale) topology optimization when multiple constraints are present: an important example being the method of moving asymptotes (MMA). We depart by noting that the aforementioned SAO algorithms are invariably formulated using strictly convex subproblems. We then numerically illustrate that strictly concave constraint functions, like those present in volumetric penalization, as recently proposed by Bruns and co-workers, may increase the difficulty of the topology optimization problem when strictly convex approximations are used in the SAO algorithm. In turn, volumetric penalization methods are of notable importance, since they seem to hold much promise for generating predominantly solid-void or discrete designs. We then argue that the nonconvex problems we study may in some instances efficiently be solved using dual SAO methods based on nonconvex (strictly concave) approximations which exhibit monotonicity with respect to the design variables. Indeed, for the topology problem resulting from SIMP-like volumetric penalization, we show explicitly that convex approximations are not necessary. Even though the volumetric penalization constraint is strictly concave, the maximum of the resulting dual subproblem still corresponds to the optimum of the original primal approximate subproblem. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Topology optimization SIMP Volumetric penalization Sequential approximate optimization (SAO) nonconvex programming Falk dual

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Simplicial Branch-and-Reduce Algorithm for Convex Programs with a Multiplicative Constraint

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2010年第2期145卷 213-233页

作者： Benson, H. P. Univ Florida Warrington Coll Business Adm Gainesville FL 32611 USA

Optimization problems that involve products of convex functions in the objective function or in the constraints arise in a variety of applications. These problems are difficult global optimization problems. During the past 15 years, however, a number of practical algorithms have been proposed for globally solving these types of problems. In this article, we present and validate a branch-and-reduce algorithm for finding a global optimal solution to a convex program that contains an additional constraint on the product of several convex functions. To globally solve this problem, the algorithm instead globally solves an equivalent master problem. At any stage of the algorithm, a disconnected set consisting of a union of simplices is constructed. This set is guaranteed to contain a portion of the boundary of the feasible region of the master problem where a global optimal solution lies. The algorithm uses a new branch-and-reduce scheme to iteratively reduce the sizes of these sets until a global optimal solution is found. Several potential computational advantages of the algorithm are explained, and a numerical example is solved.

关键词： Global optimization Multiplicative constraint nonconvex programming Product of convex functions

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Maximality Principle and General Results of Ekeland and Caristi Types without Lower Semicontinuity Assumptions in Cone Uniform Spaces with Generalized Pseudodistances

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FIXED POINT THEORY AND APPLICATIONS 2010年第1期000卷 1-35页

作者： Wlodarczyk, Kazimierz Plebaniak, Robert Univ Lodz Dept Nonlinear Anal Fac Math & Comp Sci PL-90238 Lodz Poland

Our aim is twofold: first, we want to introduce a partial quasiordering in cone uniform spaces with generalized pseudodistances for giving the general maximality principle in these spaces. Second, we want to show how this maximality principle can be used to obtain new and general results of Ekeland and Caristi types without lower semicontinuity assumptions, which was not done in the previous publications on this subject.

关键词： MAXIMAL functions CONTINUOUS functions UNIFORM spaces FIXED point theory nonconvex programming SET-valued maps MATHEMATICAL analysis

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Bisecton by Global Optimization Revisited

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2010年第3期144卷 501-510页

作者： Horst, R. Univ Trier Dept Math D-54286 Trier Germany

In partitioning methods such as branch and bound for solving global optimization problems, the so-called bisection of simplices and hyperrectangles is used since almost 40 years. Bisections are also of interest in finite-element methods. However, as far as we know, no proof has been given of the optimality of bisections with respect to other partitioning strategies. In this paper, after generalizing the current definition of partition slightly, we show that bisection is not optimal. Hybrid approaches combining different subdivision strategies with inner and outer approximation techniques can be more efficient. Even partitioning a polytope into simplices has important applications, for example in computational convexity, when one wants to find the inequality representation of a polytope with known vertices. Furthermore, a natural approach to the computation of the volume of a polytope P is to generate a simplex partition of P, since the volume of a simplex is given by a simple formula. We propose several variants of the partitioning rules and present complexity considerations. Finally, we discuss an approach for the volume computation of so-called H-polytopes, i.e., polytopes given by a system of affine inequalities. Upper bounds for the number of iterations are presented and advantages as well as drawbacks are discussed.

关键词： Global optimization nonconvex programming Mathematical programming Finite element methods Computational convexity Subdivisions Triangulation of polytopes

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The Combined Homotopy Method For Solving A Class of programming Problems With A Bounded Feasible Set

The Combined Homotopy Method For Solving A Class of Programm...

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The Third International Joint Conference on Computational Science and Optimization(第三届计算科学与优化国际大会 CSO 2010)

作者： Xiuyu Wang Qinghuai Liu Xingwu Jiang Taishan Yang School of Basic Science Changchun University of Technology Changchun 130012P.R.China Jilin Business and Technology College Changchun 130012P.R.China Mathematics School of Jilin University Changchun 130012P.R.China

In this paper, we study the following nonlinear nonconvex programming problem: {minf(x)/ε.t.g.(x)≤0,i∈M,M={1.2,...,M}. Under the condition that the feasible set is bounded and connected, but it has a point that the boundary is not regular at this point, we propose the combined homotopy method to solve this problem by constructing a new constraint function and a combined homotopy equation. The convergence of the method is proved and the existence of a smooth homotopy path from any interior point to a solution of the problem is *** method is very different from previous homotopy method. Numerical examples show that this method is feasible and effective.

关键词： nonlinear programming nonconvex programming homotopy method positively independent

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A Homotopy Method For Solving A Class of Nonlinear programming Problems

A Homotopy Method For Solving A Class of Nonlinear Programmi...

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2010 International Conference on Computer,Mechatronics, Control and Electronic Engineering (CMCE 2010)

作者： Xiuyu Wang School of Basic Science,Changchun University of Technology,Changchun,130012,P.R.China Xingwu Jiang Jilin Business and Technology College Changchun,130062,P.R.China Qinghuai Liu Institute of Applied Mathematics,Changchun University of Technology,Changchun,130012,P.R.China

In this paper,we study the following nonlinear nonconvex programming problem:min f(x),***(x) ≤0,i ∈M,M={1,2,···,m}.Under the condition that the feasible set is bounded and connected,and the feasible set does not satisfy the pseudonormal cone conditions,we propose the combined homotopy method to solve this problem by constructing new constraint functions and a combined homotopy *** convergence of the method is proved and the existence of a smooth homotopy path from any interior point to a solution of the problem is *** examples show that this method is feasible and effective.

关键词： Nonlinear programming nonconvex programming positive independence homotopy method

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