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SIAM J. on Optimization

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2006

Volume 17, Issue 4, pp. 943-1257


Stationarity Results for Generating Set Search for Linearly Constrained Optimization

Tamara G. Kolda, Robert Michael Lewis, and Virginia Torczon

SIAM J. Optim. 17, pp. 943-968 (26 pages) | Cited 13 times

Online Publication Date: November 22, 2006

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We present a new generating set search (GSS) approach for minimizing functions subject to linear constraints. GSS is a class of direct search optimization methods that includes generalized pattern search. One of our main contributions in this paper is a new condition to define the set of conforming search directions that admits several computational advantages. For continuously differentiable functions we also derive a bound relating a measure of stationarity, which is equivalent to the norm of the gradient of the objective in the unconstrained case, and a parameter used by GSS algorithms to control the lengths of the steps. With the additional assumption that the derivative is Lipschitz, we obtain a big‐$O$ bound. As a consequence of this relationship, we obtain subsequence convergence to a KKT point, even though GSS algorithms lack explicit gradient information. Numerical results indicate that the bound provides a reasonable estimate of stationarity.

Convex Approximations of Chance Constrained Programs

Arkadi Nemirovski and Alexander Shapiro

SIAM J. Optim. 17, pp. 969-996 (28 pages) | Cited 32 times

Online Publication Date: November 22, 2006

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We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its computationally tractable approximation, i.e., an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem. We construct a general class of such convex conservative approximations of the corresponding chance constrained problem. Moreover, under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent‐of‐each‐other random variables, we build a large deviation‐type approximation, referred to as “Bernstein approximation,” of the chance constrained problem. This approximation is convex and efficiently solvable. We propose a simulation‐based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and well‐known scenario approximation approaches. Finally, we extend our construction to the case of ambiguous chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly, while the chance constraint should be satisfied for every distribution given by this set.

Determinant Maximization of a Nonsymmetric Matrix with Quadratic Constraints

Serge Dégerine and Abdelhamid Zaïdi

SIAM J. Optim. 17, pp. 997-1014 (18 pages)

Online Publication Date: November 22, 2006

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This paper presents the problem of maximizing the determinant of a real $K\times K$‐matrix $B$, subject to the constraint that each row $b_k$ of $B$ satisfies $b_k^t \Gamma_kb_k \leq 1$, where $\Gamma_1, \ldots,\Gamma_K$ are $K$ given real symmetric positive definite matrices. This problem comes from a specific blind signal separation approach, but the criterion differs from approximate diagonalization criteria usually encountered in this area. Furthermore our criterion corresponds to the following nice geometrical problem: given $K$ ellipsoids in ${\rm\bf R}^K, \varepsilon_k =\{x: x^t\Gamma_kx \leq1\}, k=1, \ldots,K$, find $K$ vectors, $b_1\in \varepsilon_1, \ldots, b_K\in \varepsilon_K$, such that the volume of the parallelepiped defined by these vectors is maximum. Existence and uniqueness of the solution are discussed. An iterative algorithm, based on a relaxation technique, is proposed in order to solve this problem, and its convergence is proved under a simple sufficient condition. Some numerical experiments are performed showing the behavior of the algorithm and its comparison with Newton’s methods for nonlinear optimization.

A Proximal‐Projection Bundle Method for Lagrangian Relaxation, Including Semidefinite Programming

Krzysztof C. Kiwiel

SIAM J. Optim. 17, pp. 1015-1034 (20 pages) | Cited 6 times

Online Publication Date: December 01, 2006

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We give a proximal bundle method for minimizing a convex function $f$ over a convex set $C$. It requires evaluating $f$ and its subgradients with a fixed but possibly unknown accuracy $\epsilon>0$. Each iteration involves solving an unconstrained proximal subproblem and projecting a certain point onto $C$. The method asymptotically finds points that are $\epsilon$‐optimal. In Lagrangian relaxation of convex programs, it allows for ϵ‐accurate solutions of Lagrangian subproblems and finds ϵ‐optimal primal solutions. For semidefinite programming problems, it extends the highly successful spectral bundle method to the case of inexact eigenvalue computations.

Transposition Theorems and Qualification‐Free Optimality Conditions

Hermann Schichl and Arnold Neumaier

SIAM J. Optim. 17, pp. 1035-1055 (21 pages) | Cited 1 time

Online Publication Date: December 05, 2006

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New theorems of the alternative for polynomial constraints (based on the Positivstellensatz from real algebraic geometry) and for linear constraints (generalizing the transposition theorems of Motzkin and Tucker) are proved. Based on these, two Karush–John optimality conditions—holding without any constraint qualification—are proved for single‐ or multiobjective constrained optimization problems. The first condition applies to polynomial optimization problems only, and gives for the first time necessary and sufficient global optimality conditions for polynomial problems. The second condition applies to smooth local optimization problems and strengthens known local conditions. If some linear or concave constraints are present, the new version reduces the number of constraints for which a constraint qualification is needed to get the Kuhn–Tucker conditions.

Lyapunov Stability of Complementarity and Extended Systems

M. Kanat Camlibel, Jong‐Shi Pang, and Jinglai Shen

SIAM J. Optim. 17, pp. 1056-1101 (46 pages) | Cited 11 times

Online Publication Date: December 05, 2006

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A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear time‐invariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finite‐dimensional linear complementarity problem (LCP), whose constant vector is a linear function of the differential variable. Continuing the authors’ recent investigation of the LCS from the combined point of view of system theory and mathematical programming, this paper addresses the important system‐theoretic properties of exponential and asymptotic stability for an LCS with a C$^1$ state trajectory. The novelty of our approach lies in our employment of a quadratic Lyapunov function that involves the auxiliary algebraic variable of the LCS; when expressed in the state variable alone, the Lyapunov function is piecewise quadratic, and thus nonsmooth. The nonsmoothness feature invalidates standard stability analysis that is based on smooth Lyapunov functions. In addition to providing sufficient conditions for exponential stability, we establish a generalization of the well‐known LaSalle invariance theorem for the asymptotic stability of a smooth dynamical system to the LCS, which is intrinsically a nonsmooth system. Sufficient matrix‐theoretic copositivity conditions are introduced to facilitate the verification of the stability properties. Properly specialized, the latter conditions are satisfied by a passive‐like LCS and certain hybrid linear systems having common quadratic Lyapunov functions. We provide numerical examples to illustrate the stability results. We also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities, based on a new converse theorem for ODEs with B‐differentiable right‐hand sides. The latter theorem asserts that the existence of a “B‐differentiable Lyapunov function” is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system.

SpaseLoc: An Adaptive Subproblem Algorithm for Scalable Wireless Sensor Network Localization

Michael W. Carter, Holly H. Jin, Michael A. Saunders, and Yinyu Ye

SIAM J. Optim. 17, pp. 1102-1128 (27 pages) | Cited 11 times

Online Publication Date: December 05, 2006

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An adaptive rule‐based algorithm, SpaseLoc, is described to solve localization problems for ad hoc wireless sensor networks. A large problem is solved as a sequence of very small subproblems, each of which is solved by semidefinite programming relaxation of a geometric optimization model. The subproblems are generated according to a set of sensor/anchor selection rules. Computational results compared with existing approaches show that the SpaseLoc algorithm scales well and provides excellent localization accuracy.

Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

Akiko Yoshise

SIAM J. Optim. 17, pp. 1129-1153 (25 pages) | Cited 31 times

Online Publication Date: December 05, 2006

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We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in [L. Faybusovich, Positivity, 1 (1997), pp. 331–357] depends on the optimization theory of convex log‐barrier functions, our approach is based on the paper of Monteiro and Pang [Math. Oper. Res., 23 (1998), pp. 39–60], where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices. As an application of the results, we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects. Consequently, we show the existence of a path having the following properties: (a) The path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point of the path is a solution of the homogeneous model. (c) If the original problem is solvable, then every accumulation point of the path gives us a finite solution. (d) If the original problem is strongly infeasible, then, under the assumption of Lipschitz continuity, any accumulation point of the path gives us a finite certificate proving infeasibility.

The Lagrange Multiplier Rule for Multifunctions in Banach Spaces

Xi Yin Zheng and Kung Fu Ng

SIAM J. Optim. 17, pp. 1154-1175 (22 pages) | Cited 7 times

Online Publication Date: December 26, 2006

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We study general constrained multiobjective optimization problems with objectives being closed multifunctions in Banach spaces. In terms of the coderivatives and normal cones, we provide generalized Lagrange multiplier rules as necessary optimality conditions of the above problems. In an Asplund space setting, sharper results are presented.

New Reduction Techniques for the Group Steiner Tree Problem

Carlos Eduardo Ferreira and Fernando M. de Oliveira Filho

SIAM J. Optim. 17, pp. 1176-1188 (13 pages)

Online Publication Date: December 26, 2006

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The group Steiner tree problem consists of, given a graph $G$, a collection $\mathcal{R}$ of subsets of $V(G)$, and a positive cost $c_e$ for each edge $e$ of $G$, finding a minimum‐cost tree in $G$ that contains at least one vertex from each $R \in \mathcal{R}$. We call the sets in $\mathcal{R}$ groups. The well‐known Steiner tree problem is the special case of the group Steiner tree problem in which each set in $\mathcal{R}$ is unitary. In this paper, we present a general reduction test designed to remove group vertices, that is, vertices belonging to some group. Through the use of these tests we can conclude that a given group vertex can be considered a nonterminal and hence can be removed from its group. We also present some computational results on instances from SteinLib [T. Koch, A. Martin, and S. Voss, SteinLib: An updated library on Steiner tree problems in graphs, in Steiner Trees in Industry, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 285–325].

The Image Containment Problem and Some Classes of Polynomial Instances

Raffaele Pesenti and Franca Rinaldi

SIAM J. Optim. 17, pp. 1189-1204 (16 pages)

Online Publication Date: December 26, 2006

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The image containment problem (ICP) is a minimum cost design problem concerning the containment of particular polyhedra, called zonotopes, that are images of boxes through linear transformations. The ICP is NP‐hard. Here we study a family of nontrivial ICP instances, called worst case demand (WCD) instances. We prove that such instances can be recognized and solved in polynomial time via linear programming. Then we characterize the classes of instances that are WCD independently on the choice of the cost vector (structurally worst case demand classes (SWCD)) and we show that recognizing whether a class of instances is SWCD is a coNP‐complete problem. Finally, we describe two families of SWCD classes that are interesting from an applicative point of view: the classes defined by the incidence matrices of particular directed graphs and those defined by pre‐Leontief matrices.

The Łojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems

Jérôme Bolte, Aris Daniilidis, and Adrian Lewis

SIAM J. Optim. 17, pp. 1205-1223 (19 pages) | Cited 13 times

Online Publication Date: January 12, 2007

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Given a real‐analytic function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ and a critical point $a \in \mathbb{R}^{n}$, the Łojasiewicz inequality asserts that there exists $\theta\in\lbrack\frac{1}{2},1)$ such that the function $|f-f(a)|^{\theta}\,\Vert\nabla f\Vert^{-1}$ remains bounded around $a$. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value $+\infty$), by establishing an analogous inequality in which the derivative $\nabla f(x)$ can be replaced by any element $x^{\ast}$ of the subdifferential $\partial f(x)$ of $f$. Like its smooth version, this result provides new insights into the convergence aspects of subgradient‐type dynamical systems. Provided that the function $f$ is sufficiently regular (for instance, convex or lower‐$C^{2}$), the bounded trajectories of the corresponding subgradient dynamical system can be shown to be of finite length. Explicit estimates of the rate of convergence are also derived.

Quadratic Matrix Programming

Amir Beck

SIAM J. Optim. 17, pp. 1224-1238 (15 pages) | Cited 11 times

Online Publication Date: January 12, 2007

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We introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form $f(\boldmath $X$)={Tr}(\boldmath $X$^T \boldmath $A$ \boldmath $X$) + 2 Tr(\boldmath $B$^T \boldmath $X$) +c, \boldmath $X$ \in {\real R}^{n \times r}$. The latter formulation is termed quadratic matrix programming (QMP) of order $r$. We construct a specially devised semidefinite relaxation (SDR) and dual for the QMP problem and show that under some mild conditions strong duality holds for QMP problems with at most $r$ constraints. Using a result on the equivalence of two characterizations of the nonnegativity property of quadratic functions of the above form, we are able to compare the constructed SDR and dual problems to other known SDRs and dual formulations of the problem. An application to robust least squares problems is discussed.

Maximal Monotonicity for the Precomposition with a Linear Operator

Radu Ioan Boţ, Sorin‐Mihai Grad, and Gert Wanka

SIAM J. Optim. 17, pp. 1239-1252 (14 pages) | Cited 3 times

Online Publication Date: January 12, 2007

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We give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator $A^* \circ T \circ A$ when $A$ is a linear continuous mapping between two reflexive Banach spaces and $T$ is a maximal monotone operator. As a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive Banach space. Then we give a weak constraint qualification assuring the Brézis–Haraux‐type approximation of the range of the subdifferential of the precomposition to $A$ of a proper convex lower semicontinuous function in nonreflexive Banach spaces, extending and correcting in a special case an older result due to Riahi.

An Implementable Active‐Set Algorithm for Computing a B‐Stationary Point of a Mathematical Program with Linear Complementarity Constraints: Erratum

Masao Fukushima and Paul Tseng

SIAM J. Optim. 17, pp. 1253-1257 (5 pages) | Cited 1 time

Online Publication Date: January 22, 2007

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In [M. Fukushima and P. Tseng, SIAM J. Optim., 12 (2002), pp. 724–739], an ϵ‐active set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ϵ‐feasible set, this algorithm generates iterates whose cluster points are B‐stationary points of the problem. However, the proof has a gap and shows only that each cluster point is an M‐stationary point. We discuss this gap and show that B‐stationarity can be achieved if the algorithm is modified and an additional error bound condition holds.
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