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2007

Volume 18, Issue 4, pp. 1129-1503


Structural Topology Optimization with Eigenvalues

Wolfgang Achtziger and Michal Kočvara

SIAM J. Optim. 18, pp. 1129-1164 (36 pages) | Cited 1 time

Online Publication Date: October 10, 2007

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The paper considers different problem formulations of topology optimization of discrete or discretized structures with eigenvalues as constraints or as objective functions. We study multiple-load case formulations of minimum volume or weight, minimum compliance problems, and the problem of maximizing the minimal eigenvalue of the structure, including the effect of nonstructural mass. The paper discusses interrelations of the problems and, in particular, shows how solutions of one problem can be derived from solutions of the others. Moreover, we present equivalent reformulations as semidefinite programming problems with the property that, for the minimum volume and minimum compliance problem, each local optimizer of these problems is also a global one. This allows for the calculation of guaranteed global optimizers of the original problems through the use of modern solution techniques of semidefinite programming. For the problem of maximization of the minimum eigenvalue we show how to verify the global optimality and present an algorithm for finding a tight approximation of a globally optimal solution. Numerical examples are provided for truss structures. Both academic and larger-size examples illustrate the theoretical results achieved and demonstrate the practical use of this approach. We conclude with an extension on multiple nonstructural mass conditions.

Successive Linear Approximation Solution of Infinite-Horizon Dynamic Stochastic Programs

John R. Birge and Gongyun Zhao

SIAM J. Optim. 18, pp. 1165-1186 (22 pages)

Online Publication Date: October 10, 2007

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Models for long-term planning often lead to infinite-horizon stochastic programs that offer significant challenges for computation. Finite-horizon approximations are often used in these cases, but they may also become computationally difficult. In this paper, we directly solve for value functions of infinite-horizon stochastic programs. We show that a successive linear approximation method converges to an optimal value function for the case with convex objective, linear dynamics, and feasible continuation.

The Adaptive Convexification Algorithm: A Feasible Point Method for Semi-Infinite Programming

Christodoulos A. Floudas and Oliver Stein

SIAM J. Optim. 18, pp. 1187-1208 (22 pages) | Cited 3 times

Online Publication Date: October 10, 2007

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We present a new numerical solution method for semi-infinite optimization problems. Its main idea is to adaptively construct convex relaxations of the lower level problem, replace the relaxed lower level problems equivalently by their Karush–Kuhn–Tucker conditions, and solve the resulting mathematical programs with complementarity constraints. This approximation produces feasible iterates for the original problem. The convex relaxations are constructed with ideas from the $\alpha$BB method of global optimization. The necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic, where our algorithm already works if only one such bound is available for the problem. We show convergence of stationary points of the approximating problems to a stationary point of the original semi-infinite problem within arbitrarily given tolerances. Numerical examples from Chebyshev approximation and design centering illustrate the performance of the method.

On the Convergence of Augmented Lagrangian Methods for Constrained Global Optimization

H. Z. Luo, X. L. Sun, and D. Li

SIAM J. Optim. 18, pp. 1209-1230 (22 pages) | Cited 1 time

Online Publication Date: October 10, 2007

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In this paper, we present new convergence properties of the primal-dual method based on four types of augmented Lagrangian functions in the context of constrained global optimization. Convergence to a global optimal solution is first established for a basic primal-dual scheme under standard conditions. We then prove this convergence property for a modified augmented Lagrangian method using a safeguarding strategy without appealing to the boundedness assumption of the multiplier sequence. We further show that, under the same weaker conditions, the convergence to a global optimal solution can still be achieved by either modifying the multiplier updating rule or normalizing the multipliers in augmented Lagrangian methods.

Mathematical Programming Problems Governed by Nonlinear Elliptic PDEs

M. D. Voisei

SIAM J. Optim. 18, pp. 1231-1249 (19 pages)

Online Publication Date: October 17, 2007

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The aim of this article is to find explicit necessary conditions for local optimal pairs of problems governed by divergence-type elliptic PDEs, in terms depending on the nonlinearities involved in the cost functional and state equation. Several examples of optimization problems governed by ODEs and PDEs are presented.

Perturbation Bounds of P-Matrix Linear Complementarity Problems

Xiaojun Chen and Shuhuang Xiang

SIAM J. Optim. 18, pp. 1250-1265 (16 pages)

Online Publication Date: October 17, 2007

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We define a new fundamental constant associated with a P-matrix and show that this constant has various useful properties for the P-matrix linear complementarity problems (LCP). In particular, this constant is sharper than the Mathias–Pang constant in deriving perturbation bounds for the P-matrix LCP. Moreover, this new constant defines a measure of sensitivity of the solution of the P-matrix LCP. We examine how perturbations in the data affect the solution of the LCP and efficiency of Newton-type methods for solving the LCP.

Statistical Quasi-Newton: A New Look at Least Change

Chuanhai Liu and Scott A. Vander Wiel

SIAM J. Optim. 18, pp. 1266-1285 (20 pages)

Online Publication Date: October 24, 2007

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A new method for quasi-Newton minimization outperforms BFGS by combining least-change updates of the Hessian with step sizes estimated from a Wishart model of uncertainty. The Hessian update is in the Broyden family but uses a negative parameter, outside the convex range, that is usually regarded as the safe zone for Broyden updates. Although full Newton steps based on this update tend to be too long, excellent performance is obtained with shorter steps estimated from the Wishart model. In numerical comparisons to BFGS the new statistical quasi-Newton (SQN) algorithm typically converges with about 25% fewer iterations, functions, and gradient evaluations on the top 1/3 hardest unconstrained problems in the CUTE library. Typical improvement on the 1/3 easiest problems is about 5%. The framework used to derive SQN provides a simple way to understand differences among various Broyden updates such as BFGS and DFP and shows that these methods do not preserve accuracy of the Hessian, in a certain sense, while the new method does. In fact, BFGS, DFP, and all other updates with nonnegative Broyden parameters tend to inflate Hessian estimates, and this accounts for their observed propensity to correct eigenvalues that are too small more readily than eigenvalues that are too large. Numerical results on three new test functions validate these conclusions.

On Augmented Lagrangian Methods with General Lower-Level Constraints

R. Andreani, E. G. Birgin, J. M. Martínez, and M. L. Schuverdt

SIAM J. Optim. 18, pp. 1286-1309 (24 pages) | Cited 2 times

Online Publication Date: November 07, 2007

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Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems in which the constraints are only of the lower-level type. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the constant positive linear dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the spectral projected gradient method for solving the subproblems. Problems of this type with more than $3 \times 10^6$ variables and $ 14 \times 10^6$ constraints are solved in this way, using moderate computer time. All the codes are available at http://www.ime.usp.br/$\sim$egbirgin/tango/.

On Handling Free Variables in Interior-Point Methods for Conic Linear Optimization

Miguel F. Anjos and Samuel Burer

SIAM J. Optim. 18, pp. 1310-1325 (16 pages) | Cited 1 time

Online Publication Date: November 07, 2007

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We revisit a regularization technique of Mészáros for handling free variables within interior-point methods for conic linear optimization. We propose a simple computational strategy, supported by a global convergence analysis, for handling the regularization. Using test problems from benchmark suites and recent applications, we demonstrate that the modern code SDPT3 modified to incorporate the proposed regularization is able to achieve the same or significantly better accuracy over standard options of splitting variables, using a quadratic cone, and solving indefinite systems.

Exact Regularization of Convex Programs

Michael P. Friedlander and Paul Tseng

SIAM J. Optim. 18, pp. 1326-1350 (25 pages) | Cited 4 times

Online Publication Date: November 14, 2007

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The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a Lagrange multiplier. Moreover, the regularization parameter threshold is inversely related to the Lagrange multiplier. We use this result to generalize an exact regularization result of Ferris and Mangasarian [Appl. Math. Optim., 23 (1991), pp. 266–273] involving a linearized selection problem. We also use it to derive necessary and sufficient conditions for exact penalization, similar to those obtained by Bertsekas [Math. Programming, 9 (1975), pp. 87–99] and by Bertsekas, Nedić, and Ozdaglar [Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003]. When the regularization is not exact, we derive error bounds on the distance from the regularized solution to the original solution set. We also show that existence of a “weak sharp minimum” is in some sense close to being necessary for exact regularization. We illustrate the main result with numerical experiments on the $\ell_1$ regularization of benchmark (degenerate) linear programs and semidefinite/second-order cone programs. The experiments demonstrate the usefulness of $\ell_1$ regularization in finding sparse solutions.

Proximal Thresholding Algorithm for Minimization over Orthonormal Bases

Patrick L. Combettes and Jean-Christophe Pesquet

SIAM J. Optim. 18, pp. 1351-1376 (26 pages) | Cited 5 times

Online Publication Date: November 14, 2007

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The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing, in particular, several characterizations of such thresholders. We then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and we establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated.

On Mehrotra-Type Predictor-Corrector Algorithms

M. Salahi, J. Peng, and T. Terlaky

SIAM J. Optim. 18, pp. 1377-1397 (21 pages)

Online Publication Date: December 21, 2007

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In this paper we discuss the polynomiality of a feasible version of Mehrotra's predictor-corrector algorithm whose variants have been widely used in several interior point method (IPM)-based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms in this algorithm may lead to small steps being taken in order to keep the iterates in a large neighborhood of the central path, which is important for proving polynomial complexity properties of this method. Motivated by this example, we introduce a safeguard in Mehrotra's algorithm that keeps the iterates in the prescribed neighborhood and allows us to obtain a positive lower bound on the step size. This safeguard strategy is also used when the affine scaling direction performs poorly. We prove that the safeguarded algorithm will terminate after at most ${\cal{O}}(n^2{\rm log} (x^0)^Ts^0/\epsilon)$ iterations. By modestly modifying the corrector direction, we reduce the iteration complexity to ${\cal{O}}(n {\rm log} (x^0)^Ts^0/\epsilon).$ To ensure fast asymptotic convergence of the algorithm, we changed Mehrotra's updating scheme of the centering parameter slightly while keeping the safeguard. The new algorithms have the same order of iteration complexity as the safeguarded algorithms but enjoy superlinear convergence as well. Numerical results using the McIPM and LIPSOL software packages are reported.

Efficient Reduction of Polynomial Zero-One Optimization to the Quadratic Case

Christoph Buchheim and Giovanni Rinaldi

SIAM J. Optim. 18, pp. 1398-1413 (16 pages) | Cited 1 time

Online Publication Date: December 21, 2007

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We address the problem of optimizing a polynomial with real coefficients over binary variables. We show that a complete polyhedral description of the linearization of such a problem can be derived in a simple way from the polyhedral description of the linearization of some quadratic optimization problem. The number of variables in the latter linearization is only slightly larger than in the former. If polynomial constraints are present in the original problem, then their linearized counterparts carry over to the linearized quadratic problem unchanged. If the original problem formulation does not contain any constraints, we obtain a reduction to unconstrained quadratic zero-one optimization, which is equivalent to the well-studied max-cut problem. The separation problem for general unconstrained polynomial zero-one optimization thus reduces to the separation problem for the cut polytope. This allows us to transfer the entire knowledge gained for the latter polytope by intensive research and, in particular, the sophisticated separation techniques that have been developed. We report preliminary experimental results obtained with a straightforward implementation of this approach.

A High-Order Path-Following Method for Locating the Least 2-Norm Solution of Monotone LCPs

Anhua Lin

SIAM J. Optim. 18, pp. 1414-1435 (22 pages) | Cited 1 time

Online Publication Date: January 16, 2008

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A high-order path-following method is proposed for finding the least 2-norm solution of a monotone LCP. This method follows the regularized central paths introduced in [Y. B. Zhao and D. Li, SIAM J. Control Optim., 40 (2001), pp. 898–924]. By using the analyticity of these paths, we showed the global convergence under the assumption that the LCP has at least one solution and the superlinear rate of local convergence under the assumption that the least 2-norm solution is maximally complementary.

Explicit Reformulations for Robust Optimization Problems with General Uncertainty Sets

Igor Averbakh and Yun-Bin Zhao

SIAM J. Optim. 18, pp. 1436-1466 (31 pages) | Cited 1 time

Online Publication Date: January 16, 2008

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We consider a rather general class of mathematical programming problems with data uncertainty, where the uncertainty set is represented by a system of convex inequalities. We prove that the robust counterparts of this class of problems can be reformulated equivalently as finite and explicit optimization problems. Moreover, we develop simplified reformulations for problems with uncertainty sets defined by convex homogeneous functions. Our results provide a unified treatment of many situations that have been investigated in the literature and are applicable to a wider range of problems and more complicated uncertainty sets than those considered before. The analysis in this paper makes it possible to use existing continuous optimization algorithms to solve more complicated robust optimization problems. The analysis also shows how the structure of the resulting reformulation of the robust counterpart depends both on the structure of the original nominal optimization problem and on the structure of the uncertainty set.

A Method of Centers with Approximate Subgradient Linearizations for Nonsmooth Convex Optimization

Krzysztof C. Kiwiel

SIAM J. Optim. 18, pp. 1467-1489 (23 pages) | Cited 1 time

Online Publication Date: January 16, 2008

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We give a proximal bundle method for constrained convex optimization. It requires only evaluating the problem functions and their subgradients with an unknown accuracy $\epsilon$. Employing a combination of the classic method of centers' improvement function with an exact penalty function, it does not need a feasible starting point. It asymptotically finds points with at least $\epsilon$-optimal objective values that are $\epsilon$-feasible. When applied to the solution of linear programming problems via column generation, it allows for $\epsilon$-accurate solutions of column generation subproblems.

Improved Approximation Algorithms for Weighted Hypergraph Embedding in a Cycle

Hann-Jang Ho and SingLing Lee

SIAM J. Optim. 18, pp. 1490-1500 (11 pages)

Online Publication Date: January 16, 2008

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We consider the problem of embedding weighted hyperedges of a hypergraph as paths in a cycle on the same number of vertices, such that the maximum congestion of any physical link of the cycle is minimized. The problem, called weighted hypergraph embedding in a cycle (WHEC), is known to be NP-complete even when each hyperedge is unweighted or each weighted hyperedge contains exactly two vertices. In this paper, we propose an improved rounding algorithm for the WHEC problem to provide a solution with an approximation bound of $1.5(opt+w_{max})$, where $opt$ represents the optimal value of the problem and $w_{max}$ denotes the largest weight of hyperedges. For any fixed $\varepsilon > 0$, we also present a polynomial time algorithm to provide an embedding whose congestion is at most $(1.5+\varepsilon)$ times the optimum. This improves previous results for the general WHEC problem.

Erratum: Mesh Adaptive Direct Search Algorithms for Constrained Optimization

Charles Audet, A. L. Custódio, and J. E. Dennis,, Jr.

SIAM J. Optim. 18, pp. 1501-1503 (3 pages) | Cited 3 times

Online Publication Date: January 16, 2008

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In [SIAM J. Optim., 17 (2006), pp. 188–217] Audet and Dennis proposed the class of mesh adaptive direct search (MADS) algorithms for minimization of a nonsmooth function under general nonsmooth constraints. The notation used in the paper evolved since the preliminary versions, and, unfortunately, even though the statement of Proposition 4.2 is correct, it is not compatible with the final notation. The purpose of this note is to show that the proposition is valid.
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