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SIAM J. on Optimization

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2005

Volume 16, Issue 2, pp. 311-628


Strong CHIP for Infinite System of Closed Convex Sets in Normed Linear Spaces

Chong Li and K. F. Ng

SIAM J. Optim. 16, pp. 311-340 (30 pages) | Cited 7 times

Online Publication Date: July 28, 2006

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For a general (possibly infinite) system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property. One set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interior-point conditions together with appropriate continuity of the associated set-valued function on the (topologized) index set I. In the special case when I is finite and X is finite dimensional, one of these results reduces to a classical result of Rockafellar.

A Filter-Trust-Region Method for Unconstrained Optimization

Nick I. M. Gould, Caroline Sainvitu, and Philippe L. Toint

SIAM J. Optim. 16, pp. 341-357 (17 pages) | Cited 24 times

Online Publication Date: July 28, 2006

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A new filter-trust-region algorithm for solving unconstrained nonlinear optimization problems is introduced. Based on the filter technique introduced by Fletcher and Leyffer, it extends an existing technique of Gould, Leyffer, and Toint [SIAM J. Optim., 15 (2004), pp. 17--38] for nonlinear equations and nonlinear least-squares to the fully general unconstrained optimization problem. The new algorithm is shown to be globally convergent to at least one second-order critical point, and numerical experiments indicate that it is very competitive with more classical trust-region algorithms.

Adjusted Sublevel Sets, Normal Operator, and Quasi-convex Programming

D. Aussel and N. Hadjisavvas

SIAM J. Optim. 16, pp. 358-367 (10 pages) | Cited 7 times

Online Publication Date: July 28, 2006

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A new notion of "adjusted sublevel set" of a function is introduced and studied. These sets lie between the sublevel and strict sublevel sets of the function. In contrast to the normal operators to sublevel or strict sublevel sets that were studied in the literature so far, the normal operator to the adjusted sublevel sets is both quasi-monotone and, in the case of quasi-convex functions, cone upper-semicontinuous. This makes this new notion appropriate for all kinds of quasi-convex functions and, in particular, for quasi-convex functions whose graph presents a "flat part." Application is given to quasi-convex optimization through the study of an associated variational inequality problem.

Minimizing within Convex Bodies Using a Convex Hull Method

Thomas Lachand-Robert and Édouard Oudet

SIAM J. Optim. 16, pp. 368-379 (12 pages) | Cited 14 times

Online Publication Date: July 28, 2006

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We present numerical methods to solve optimization problems on the space of convex functions or among convex bodies. Hence convexity is a constraint on the admissible objects, whereas the functionals are not required to be convex. To deal with this, our method mixes geometrical and numerical algorithms.
We give several applications arising from classical problems in geometry and analysis: Alexandrov's problem of finding a convex body of prescribed surface function; Cheeger's problem of a subdomain minimizing the ratio surface area on volume; Newton's problem of the body of minimal resistance.
In particular for the latter application, the minimizers are still unknown, except in some particular classes. We give approximate solutions better than the theoretical known ones, hence demonstrating that the minimizers do not belong to these classes.

On Lovász--Schrijver Lift-and-Project Procedures on the Dantzig--Fulkerson--Johnson Relaxation of the TSP

Kevin K. H. Cheung

SIAM J. Optim. 16, pp. 380-399 (20 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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We study the Lovász--Schrijver lift-and-project procedure $N_+$ on the linear relaxation of the Dantzig--Fulkerson--Johnson formulation of the traveling salesman problem (TSP). A long standing conjecture states that the integrality gap of this relaxation is $\frac43$ in the case of metric costs. In this paper, we show that the $N_+$-rank of 2-matching inequalities relative to this relaxation can be arbitrarily high and obtain as a corollary that even after applying $N_+$ to the relaxation a fixed number of times, the integrality gap of the resulting relaxation is at least $\frac43$.

An O$(\sqrtn L)$ Iteration Primal-dual Path-following Method, Based on Wide Neighborhoods and Large Updates, for Monotone LCP

Wenbao Ai and Shuzhong Zhang

SIAM J. Optim. 16, pp. 400-417 (18 pages) | Cited 12 times

Online Publication Date: July 28, 2006

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In this paper we propose a new class of primal-dual path-following interior point algorithms for solving monotone linear complementarity problems. At each iteration, the method would select a target on the central path with a large update from the current iterate, and then the Newton method is used to get the search directions, followed by adaptively choosing the step sizes, which are, e.g., the largest possible steps before leaving a neighborhood that is as wide as the given ${\cal N}^-_{\infty}$ neighborhood. The only deviation from the classical approach is that we treat the classical Newton direction as the sum of two other directions, corresponding to, respectively, the negative part and the positive part of the right-hand side. We show that if these two directions are equipped with different and appropriate step sizes, then the method enjoys the low iteration bound of $O(\sqrt{n}\log L)$, where $n$ is the dimension of the problem and $L=\frac{(x^0)^Ts^0}{\ep}$ with $\ep$ the required precision and $(x^0,s^0)$ the initial interior solution. For a predictor-corrector variant of the method, we further prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of $1-1/O(\sqrt{n})$. Additionally, if the problem has a strict complementary solution, then the predictor steps converge Q-quadratically.

Validated Linear Relaxations and Preprocessing: Some Experiments

R. Baker Kearfott and Siriporn Hongthong

SIAM J. Optim. 16, pp. 418-433 (16 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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Based on work originating in the early 1970s, a number of recent global optimization algorithms have relied on replacing an original nonconvex nonlinear program by convex or linear relaxations. Such linear relaxations can be generated automatically through an automatic differentiation process. This process decomposes the objective and constraints (if any) into convex and nonconvex unary and binary operations. The convex operations can be approximated arbitrarily well by appending additional constraints, while the domain must somehow be subdivided (in an overall branch-and-bound process or in some other local process) to handle nonconvex constraints. In general, a problem can be hard if even a single nonconvex term appears. However, certain nonconvex terms lead to easier-to-solve problems than others. Recently, Neumaier, Lebbah, Michel, ourselves, and others have paved the way to utilizing such techniques in a validated context.
In this paper, we present a symbolic preprocessing step that provides a measure of the intrinsic difficulty of a problem. Based on this step, one of two methods can be chosen to relax nonconvex terms. This preprocessing step is similar to a method previously proposed by Epperly and Pistikopoulos [J. Global Optim., 11 (1997), pp. 287--311] for determining subspaces in which to branch, but we present it from a different point of view that is amenable to simplification of the problem presented to the linear programming solver, and within a validated context. Besides an illustrative example, we have implemented general relaxations in a validated context, as well as the preprocessing technique, and we present experiments on a standard test set. Finally, we present conclusions.

An Efficient Approximate Algorithm for the 1-Median Problem in Metric Spaces

D. Cantone, G. Cincotti, A. Ferro, and A. Pulvirenti

SIAM J. Optim. 16, pp. 434-451 (18 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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We propose a simple and natural linear randomized algorithm for the approximate 1-median selection problem in metric spaces. The 1-median of a finite subset S of a metric space is the element of S which minimizes the average distance from the remaining points in S. This problem is extremely important in most applications using clustering of metric spaces, but also in connection with several algorithms in bioinformatics. The only linear approximation algorithm for the 1-median problem, which provably works in any metric space without going through any Euclidean space, has been proposed by Indyk in [Proceedings of the 31st Annual ACM Symposium on Theory of Computing, Atlanta, 1999, pp. 428--432]. However, Indyk's algorithm, which is based on sufficiently large sampling, turns out not to be a practical solution. The same holds true even for its heuristic variants which use samplings of smaller size. The algorithm we propose has a simple and efficient implementation, which performs better than Indyk's algorithm in practice. On the other hand, while the performance of Indyk's algorithm is guaranteed by an approximation factor, in the case of our algorithm we are only able to produce experimental evidence of its precision. Extensive experimentation has been performed on both synthetic and real input datasets. Synthetic datasets were generated with uniform and skewed distributions, using various metrics. Real datasets have been extrapolated from real world official databases available on the web. Successful results of the proposed algorithm are reported for several applications in bioinformatics and various classes of approximate search queries.

An Algebraic Condition Equivalent to Strong Stability of Stationary Solutions of Nonlinear Positive Semidefinite Programs

Toshihiro Matsumoto

SIAM J. Optim. 16, pp. 452-470 (19 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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This paper addresses strong stability, in the sense of Kojima, of stationary solutions of nonlinear positive semidefinite programs (NSDP). First, we give a characterization of stability from the point of view of one-to-one maps under an LICQ condition generalized to these programs. Second, under the same condition we construct a method for NSDP that is analogous to Kojima's method for classical nonlinear programs (NLP) treated in his famous paper. From this construction we make clear the essential difference between NSDP and NLP, and we deduce an algebraic condition equivalent to strong stability for those NSDP to which there does not exist this difference.

On the Convergence of Successive Linear-Quadratic Programming Algorithms

Richard H. Byrd, Nicholas I. M. Gould, Jorge Nocedal, and Richard A. Waltz

SIAM J. Optim. 16, pp. 471-489 (19 pages) | Cited 16 times

Online Publication Date: July 28, 2006

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The global convergence properties of a class of penalty methods for nonlinear programming are analyzed. These methods include successive linear programming approaches and, more specifically, the successive linear-quadratic programming approach presented by Byrd et al. [Math. Program., 100 (2004), pp. 27--48]. Every iteration requires the solution of two trust-region subproblems involving piecewise linear and quadratic models, respectively. It is shown that, for a fixed penalty parameter, the sequence of iterates approaches stationarity of the penalty function. A procedure for dynamically adjusting the penalty parameter is described, and global convergence results for it are established.

Semidefinite Approximations for Global Unconstrained Polynomial Optimization

Dorina Jibetean and Monique Laurent

SIAM J. Optim. 16, pp. 490-514 (25 pages) | Cited 12 times

Online Publication Date: July 28, 2006

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We consider the problem of minimizing a polynomial function on ${\mathbb R}^n$, known to be hard even for degree $4$ polynomials. Therefore approximation algorithms are of interest. Lasserre [SIAM J. Optim., 11 (2001), pp. 796--817] and Parrilo [Math. Program., 96 (2003), pp. 293--320] have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations. We propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming. The method is applied to several examples.

Second-Order Behavior of Pattern Search

Mark A. Abramson

SIAM J. Optim. 16, pp. 515-530 (16 pages) | Cited 7 times

Online Publication Date: July 28, 2006

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Previous analyses of pattern search algorithms for unconstrained and linearly constrained minimization have focused on proving convergence of a subsequence of iterates to a limit point satisfying either directional or first-order necessary conditions for optimality, depending on the smoothness of the objective function in a neighborhood of the limit point. Even though pattern search methods require no derivative information, we are able to prove some limited directional second-order results. Although not as strong as classical second-order necessary conditions, these results are stronger than the first-order conditions that many gradient-based methods satisfy. Under fairly mild conditions, we can eliminate from consideration all strict local maximizers and an entire class of saddle points.

Convergence of the Iterates of Descent Methods for Analytic Cost Functions

P. A. Absil, R. Mahony, and B. Andrews

SIAM J. Optim. 16, pp. 531-547 (17 pages) | Cited 16 times

Online Publication Date: July 28, 2006

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In the early eighties Lojasiewicz [in Seminari di Geometria 1982-1983, Università di Bologna, Istituto di Geometria, Dipartimento di Matematica, 1984, pp. 115--117] proved that a bounded solution of a gradient flow for an analytic cost function converges to a well-defined limit point. In this paper, we show that the iterates of numerical descent algorithms, for an analytic cost function, share this convergence property if they satisfy certain natural descent conditions. The results obtained are applicable to a broad class of optimization schemes and strengthen classical "weak convergence" results for descent methods to "strong limit-point convergence" for a large class of cost functions of practical interest. The result does not require that the cost has isolated critical points and requires no assumptions on the convexity of the cost nor any nondegeneracy conditions on the Hessian of the cost at critical points.

A Note on Trust-Region Radius Update

Jérome M. B. Walmag and Éric J. M. Delhez

SIAM J. Optim. 16, pp. 548-562 (15 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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In classical trust-region optimization algorithms, the radius of the trust region is reduced, kept constant, or enlarged after, respectively, unsuccessful, successful, and very successful iterations. We propose here to refine the empirical rules used for this update by the definition of a new set of iterations that we call "too successful iterations." At such iterations, a large reduction of the objective function is obtained despite a crude local approximation of the objective function; the trust region is thus kept nearly constant instead of being enlarged.
The new update rules preserve the strong convergence property of traditional trust-region methods. They can also be generalized to define a self-adaptive trust-region algorithm along the lines introduced by Hei [J. Comput. Math., 21 (2003), pp. 229-236].
Numerical experiments carried out on 70 unconstrained problems from the CUTEr collection demonstrate the positive impact of the modified update strategy on the efficiency and robustness of quasi-Newton variants of a trust-region solver, when BFGS or SR1 updates of the approximation of the Hessian matrix are carried at all iterations.

Revisiting Asynchronous Parallel Pattern Search for Nonlinear Optimization

Tamara G. Kolda

SIAM J. Optim. 16, pp. 563-586 (24 pages) | Cited 22 times

Online Publication Date: July 28, 2006

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We present a new asynchronous parallel pattern search (APPS) method which is different from that developed previously by Hough, Kolda, and Torczon. APPS efficiently uses parallel and distributed computing platforms to solve science and engineering design optimization problems where derivatives are unavailable and cannot be approximated. The original APPS was designed to be fault-tolerant as well as asynchronous and was based on a peer-to-peer design. Each process was in charge of a single, fixed search direction. Our new version is based instead on a manager-worker paradigm. Though less fault-tolerant, the resulting algorithm is more flexible in its use of distributed computing resources. We further describe how to incorporate a zero-order sufficient decrease condition and handle bound constraints. Convergence theory for all situations (unconstrained and bound constrained as well as simple and sufficient decrease) is developed. We close with a discussion of how the new APPS will better facilitate the future incorporation of linear and nonlinear constraints.

A two-sided relaxation scheme for Mathematical Programs with Equilibrium Constraints

Victor DeMiguel, Michael P. Friedlander, Francisco J. Nogales, and Stefan Scholtes

SIAM J. Optim. 16, pp. 587-609 (23 pages) | Cited 20 times

Online Publication Date: July 28, 2006

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We propose a relaxation scheme for mathematical programs with equilibrium constraints (MPECs). In contrast to previous approaches, our relaxation is two-sided: both the complementarity and the nonnegativity constraints are relaxed. The proposed relaxation update rule guarantees (under certain conditions) that the sequence of relaxed subproblems will maintain a strictly feasible interior---even in the limit. We show how the relaxation scheme can be used in combination with a standard interior-point method to achieve superlinear convergence. Numerical results on the MacMPEC test problem set demonstrate the fast local convergence properties of the approach.

Sum of Squares Approximation of Polynomials, Nonnegative on a Real Algebraic Set

Jean B. Lasserre

SIAM J. Optim. 16, pp. 610-628 (19 pages) | Cited 7 times

Online Publication Date: July 28, 2006

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Wih every real polynomial $f$, we associate a family $\{f_{\epsilon r}\}_{\epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $\epsilon>0,r\in\N$, and such that $\Vert f-f_{\epsilon r}\Vert_1\to 0$ as $\epsilon\to 0$.
Let $V\subset \R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $\epsilon>0$, there exist nonnegative scalars $\{\lambda_j(\epsilon)\}_{j\in J}$ such that, for all $r$ sufficiently large, \[f_{\epsilon r}+\sum_{j\in J} \lambda_j(\epsilon)\,g_j^2\quad\mbox{is a sum of squares.}\] This representation is an obvious certificate of nonnegativity of $f_{\epsilon r}$ on $V$, and very specific in terms of the $g_j$ that define the set $V$. In particular, it is valid with no assumption on $V$. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programming relaxations to approximate the global minimum of $f$ on a real algebraic set $V$, or a semialgebraic set $\K$, and again, with no assumption on $V$ or $\K$.
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