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SIAM J. on Optimization

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2006

Volume 17, Issue 1, pp. 1-310


Solving Matrix Inequalities whose Unknowns are Matrices

Juan F. Camino, J. William Helton, and Robert E. Skelton

SIAM J. Optim. 17, pp. 1-36 (36 pages)

Online Publication Date: July 28, 2006

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This paper provides algorithms for numerical solution of convex matrix inequalities in which the variables naturally appear as matrices. This includes, for instance, many systems and control problems. To use these algorithms, no knowledge of linear matrix inequalities is required. However, as tools, they preserve many advantages of the linear matrix inequality framework. Our method has two components: (1) a numerical algorithm that solves a large class of matrix optimization problems and (2) a symbolic "convexity checker" that automatically provides a region which, if convex, guarantees that the solution from (1) is a global optimum on that region. The algorithms are partly numerical and partly symbolic and since they aim at exploiting the matrix structure of the unknowns, the symbolic part requires the development of new computer techniques for treating noncommutative algebra.

Constructing Generalized Mean Functions Using Convex Functions with Regularity Conditions

Yun-Bin Zhao, Shu-Cherng Fang, and Duan Li

SIAM J. Optim. 17, pp. 37-51 (15 pages)

Online Publication Date: July 28, 2006

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The generalized mean function has been widely used in convex analysis and mathematical programming. This paper studies a further generalization of such a function. A necessary and sufficient condition is obtained for the convexity of a generalized function. Additional sufficient conditions that can be easily checked are derived for the purpose of identifying some classes of functions which guarantee the convexity of the generalized functions. We show that some new classes of convex functions with certain regularity (such as S*-regularity) can be used as building blocks to construct such generalized functions.

Interior Methods for Mathematical Programs with Complementarity Constraints

Sven Leyffer, Gabriel López-Calva, and Jorge Nocedal

SIAM J. Optim. 17, pp. 52-77 (26 pages) | Cited 33 times

Online Publication Date: July 28, 2006

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This paper studies theoretical and practical properties of interior-penalty methods for mathematical programs with complementarity constraints. A framework for implementing these methods is presented, and the need for adaptive penalty update strategies is motivated with examples. The algorithm is shown to be globally convergent to strongly stationary points, under standard assumptions. These results are then extended to an interior-relaxation approach. Superlinear convergence to strongly stationary points is also established. Two strategies for updating the penalty parameter are proposed, and their efficiency and robustness are studied on an extensive collection of test problems.

Discrete Monotonic Optimization with Application to a Discrete Location Problem

Hoang Tuy, Michel Minoux, and N. T. Hoai-Phuong

SIAM J. Optim. 17, pp. 78-97 (20 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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A general discrete optimization problem is investigated that includes integer polynomial programs as special cases. To exploit the discrete monotonic structure of these problems, a special class of cuts called monotonicity cuts are developed and then adjusted according to a suitable procedure to accommodate discrete requirements. As illustration, the method is applied to solve a discrete location problem which is also a variant of the well known engineering problem of design centering. Computational results are reported for instances of the latter problem with up to 100 variables and 500 constraints.

On the Solution of the Tikhonov Regularization of the Total Least Squares Problem

Amir Beck and Aharon Ben-Tal

SIAM J. Optim. 17, pp. 98-118 (21 pages) | Cited 11 times

Online Publication Date: July 28, 2006

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Total least squares (TLS) is a method for treating an overdetermined system of linear equations ${\bf A} {\bf x} \approx {\bf b}$, where both the matrix ${\bf A}$ and the vector ${\bf b}$ are contaminated by noise. Tikhonov regularization of the TLS (TRTLS) leads to an optimization problem of minimizing the sum of fractional quadratic and quadratic functions. As such, the problem is nonconvex. We show how to reduce the problem to a single variable minimization of a function ${\mathcal{G}}$ over a closed interval. Computing a value and a derivative of ${\mathcal{G}}$ consists of solving a single trust region subproblem. For the special case of regularization with a squared Euclidean norm we show that ${\mathcal{G}}$ is unimodal and provide an alternative algorithm, which requires only one spectral decomposition. A numerical example is given to illustrate the effectiveness of our method.

Constraint Reduction for Linear Programs with Many Inequality Constraints

André L. Tits, P. A. Absil, and William P. Woessner

SIAM J. Optim. 17, pp. 119-146 (28 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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Consider solving a linear program in standard form where the constraint matrix $A$ is $m \times n$, with $n \gg m \gg 1$. Such problems arise, for example, as the result of finely discretizing a semi-infinite program. The cost per iteration of typical primal-dual interior-point methods on such problems is $O(m^2n)$. We propose to reduce that cost by replacing the normal equation matrix, $AD^2A^{\T}$, where $D$ is a diagonal matrix, with a "reduced" version (of same dimension), $A_QD_Q^2A_Q^{\T}$, where $Q$ is an index set including the indices of $M$ most nearly active (or most violated) dual constraints at the current iterate, with $M\geq m$ a prescribed integer. This can result in a speedup of close to $n/|Q|$ at each iteration. Promising numerical results are reported for constraint-reduced versions of a dual-feasible affine-scaling algorithm and of Mehrotra's predictor-corrector method [S. Mehrotra, it SIAM J. Optim., 2 (1992), pp. 575-601]. In particular, while it could be expected that neglecting a large portion of the constraints, especially at early iterations, may result in a significant deterioration of the search direction, it appears that the total number of iterations typically remains essentially constant as the size of the reduced constraint set is decreased down to some threshold. In some cases this threshold is a small fraction of the total set. In the case of the affine-scaling algorithm, global convergence and local quadratic convergence are proved.

On the Stability of Convex-valued Mappings and Their Relative Boundary and Extreme Points Set Mappings

Miguel A. Goberna, Maxim I. Todorov, and Virginia N. Vera de Serio

SIAM J. Optim. 17, pp. 147-158 (12 pages)

Online Publication Date: July 28, 2006

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This paper deals with the transmission of the main stability properties (lower and upper semicontinuity in Berge sense, and closedness) from a given closed--convex-valued mapping to its corresponding relative boundary and extreme point set mappings, and vice versa. The domain of the mappings considered in this paper are locally metrizable spaces and the images range on Euclidean spaces. Important examples of the class of mappings considered in this paper are the feasible set mapping and the optimal set mapping of convex optimization problems, for which the space of parameters is the result of perturbing a given nominal problem.

Path-following Methods for a Class of Constrained Minimization Problems in Function Space

Michael Hintermüller and Karl Kunisch

SIAM J. Optim. 17, pp. 159-187 (29 pages) | Cited 26 times

Online Publication Date: July 28, 2006

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Path-following methods for primal-dual active set strategies requiring a regularization parameter are introduced. Existence of a primal-dual path and its differentiability properties are analyzed. Monotonicity and convexity of the primal-dual path value function are investigated. Both feasible and infeasible approximations are considered. Numerical path-following strategies are developed and their efficiency is demonstrated by means of examples.

Mesh Adaptive Direct Search Algorithms for Constrained Optimization

Charles Audet and J. E. Dennis, Jr.

SIAM J. Optim. 17, pp. 188-217 (30 pages) | Cited 84 times

Online Publication Date: July 28, 2006

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This paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available. We introduce the mesh adaptive direct search (MADS) class of algorithms which extends the generalized pattern search (GPS) class by allowing local exploration, called polling, in an asymptotically dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained.
The main GPS convergence result is to identify limit points $\hat{x}$, where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions span the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for GPS. The main result of this paper is that the general MADS framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the Clarke derivatives are nonnegative.
We propose an instance of MADS for which the refining directions are dense in the hypertangent cone at $\hat{x}$ with probability 1 whenever the iterates associated with the refining directions converge to a single $\hat{x}$. The instance of MADS is compared to versions of GPS on some test problems. We also illustrate the limitation of our results with examples.
An erratum to this article has been appended at the end of the pdf file.

Sums of Squares and Semidefinite Program Relaxations for Polynomial Optimization Problems with Structured Sparsity

Hayato Waki, Sunyoung Kim, Masakazu Kojima, and Masakazu Muramatsu

SIAM J. Optim. 17, pp. 218-242 (25 pages) | Cited 12 times

Online Publication Date: July 28, 2006

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Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of the supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite program (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations.

Generalized Levitin--Polyak Well-Posedness in Constrained Optimization

X. X. Huang and X. Q. Yang

SIAM J. Optim. 17, pp. 243-258 (16 pages)

Online Publication Date: July 28, 2006

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In this paper, we consider Levitin--Polyak-type well-posedness for a general constrained optimization problem. We introduce generalized Levitin--Polyak well-posedness and strongly generalized Levitin--Polyak well-posedness. Necessary and sufficient conditions for these types of well-posedness are given. Relations among these types of well-posedness are investigated. Finally, we consider convergence of a class of penalty methods and a class of augmented Lagrangian methods under the assumption of strongly generalized Levitin--Polyak well-posedness.

Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints

Roger Fletcher, Sven Leyffer, Danny Ralph, and Stefan Scholtes

SIAM J. Optim. 17, pp. 259-286 (28 pages) | Cited 9 times

Online Publication Date: July 28, 2006

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Recently, nonlinear programming solvers have been used to solve a range of mathematical programs with equilibrium constraints (MPECs). In particular, sequential quadratic programming (SQP) methods have been very successful. This paper examines the local convergence properties of SQP methods applied to MPECs. SQP is shown to converge superlinearly under reasonable assumptions near a strongly stationary point. A number of examples are presented that show that some of the assumptions are difficult to relax.

An Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex Quadratic Programming

Zhaosong Lu, Renato D. C. Monteiro, and Jerome W. O'Neal

SIAM J. Optim. 17, pp. 287-310 (24 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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In this paper we develop a long-step primal-dual infeasible path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. We propose a new linear system, which we refer to as the augmented normal equation (ANE), to determine the primal-dual search directions. Since the condition number of the ANE coefficient matrix may become large for degenerate CQP problems, we use a maximum weight basis preconditioner introduced in [A. R. L. Oliveira and D. C. Sorensen, Linear Algebra Appl., 394 (2005), pp. 1-24; M. G. C. Resende and G. Veiga, SIAM J. Optim., 3 (1993), pp. 516-537; P. Vaida, Solving Linear Equations with Symmetric Diagonally Dominant Matrices by Constructing Good Preconditioners, Tech. report, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, 1990] to precondition this matrix. Using a result obtained in [R. D. C. Monteiro, J. W. O'Neal, and T. Tsuchiya, SIAM J. Optim., 15 (2004), pp. 96-100], we establish a uniform bound, depending only on the CQP data, forthe number of iterations needed by the iterative linear solver to obtain a sufficiently accurate solution to the ANE. Since the iterative linear solver can generate only an approximate solution to the ANE, this solution does not yield a primal-dual search direction satisfying all equations of the primal-dual Newton system. We propose a way to compute an inexact primal-dual search direction so that the equation corresponding to the primal residual is satisfied exactly, while the one corresponding to the dual residual contains a manageable error which allows us to establish a polynomial bound on the number of iterations of our method.
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