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SIAM J. on Optimization

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2005

Volume 15, Issue 4, pp. 953-1261


Proximal Methods in Vector Optimization

Henri Bonnel, Alfredo Noel Iusem, and Benar Fux Svaiter

SIAM J. Optim. 15, pp. 953-970 (18 pages) | Cited 11 times

Online Publication Date: July 28, 2006

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We consider the vector optimization problem of finding weakly efficient points for maps from a Hilbert space X to a Banach space Y with respect to the partial order induced by a closed, convex, and pointed cone $C\subset Y$ with a nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization. In this extension, the subproblems consist of finding weakly efficient points for suitable regularizations of the original map. We present both an exact and an inexact version, in which the subproblems are solved only approximately, within a constant relative tolerance. In both cases, we prove weak convergence of the generated sequence to a weakly efficient point, assuming convexity of the map with respect to C and C-completeness of the initial section. In cases where this last assumption fails, we still establish that the generating sequence is, in a suitable sense, a minimizing one. We also exhibit a particular instance of the algorithm for which, under a mild hypothesis on C, the weak limit of the generated sequence is an efficient, rather than a weakly efficient, point.

Solving Multicommodity Flow Problems by an Approximation Scheme

Jorge Villavicencio

SIAM J. Optim. 15, pp. 971-986 (16 pages)

Online Publication Date: July 28, 2006

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We study an approximation scheme to solve minimum cost multicommodity flow problems to a relative accuracy of $\varepsilon\in (0,1]$. The proposed scheme, which we shall call Algorithm ${\cal A}$, is a bisection-based procedure, so it maintains an interval that contains the optimal value. At each iteration, Algorithm ${\cal A}$ defines a block angular sharing problem that is solved to a relative accuracy of $O(\varepsilon)$. The computed solution defines the current approximation to an optimal solution and it is used by Algorithm ${\cal A}$ to throw away half of the current interval. It is shown that when Algorithm ${\cal A}$ no longer shrinks the current interval, the current solution solves the minimum cost multicommodity flow problem to the given accuracy $\varepsilon$. To compute approximate solutions to the sharing problems that Algorithm ${\cal A}$ defines, we propose using Algorithm ${\cal L}$ from [J. Villavicencio and M. D. Grigoriadis, Approximate Lagrangian decomposition with a modified Karmarkar logarithmic potential, in Network Optimization, Lecture Notes in Econom. and Math. Systems 450, Springer, Berlin, 1997, pp. 471--485], but we replace its fixed small step size by one computed using inexact line searches. This modification allows large steps in Algorithm ${\cal L}$. We show that the coordination complexities of Algorithm ${\cal L}$ and Algorithm ${\cal L}$ with inexact line searches are the same if these algorithms are applied to the block angular sharing problems defined by Algorithm ${\cal A}$. This result is also true for block angular sharing problems with convex blocks and with coupling constraints given by nonnegative linear functions.

Scalarizing Functions for Generating the Weakly Efficient Solution Set in Convex Multiobjective Problems

Dinh The Luc, Thai Quynh Phong, and Michel Volle

SIAM J. Optim. 15, pp. 987-1001 (15 pages) | Cited 8 times

Online Publication Date: July 28, 2006

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A method that uses normal directions of convex sets is proposed to construct a sequence of scalarizing functions which generates all weakly efficient solutions of a convex multiobjective optimization problem. The convergence of the method is proven and some small-scale numerical examples are given.

Space Mapping for Optimal Control of Partial Differential Equations

Michael Hintermüller and Luís N. Vicente

SIAM J. Optim. 15, pp. 1002-1025 (24 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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Solving optimal control problems for nonlinear partial differential equations represents a significant numerical challenge due to the tremendous size and possible model difficulties (e.g., nonlinearities) of the discretized problems. In this paper, a novel space-mapping technique for solving the aforementioned problem class is introduced, analyzed, and tested. The advantage of the space-mapping approach compared to classical multigrid techniques lies in the flexibility of not only using grid coarsening as a model reduction but also employing (perhaps less nonlinear) surrogates. The space mapping is based on a regularization approach which, in contrast to other space-mapping techniques, results in a smooth mapping and, thus, avoids certain irregular situations at kinks. A new Broyden update formula for the sensitivities of the space map is also introduced. This quasi-Newton update is motivated by the usual secant condition combined with a secant condition resulting from differentiating the space-mapping surrogate. The overall algorithm employs a trust-region framework for global convergence. Issues involved in the computations are highlighted, and a report on a few illustrative numerical tests is given.

Perturbation Analysis of Error Bounds for Systems of Conic Linear Inequalities in Banach Spaces

Xi Yin Zheng and Kung Fu Ng

SIAM J. Optim. 15, pp. 1026-1041 (16 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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For a general (constrained or unconstrained) system of conic inequalities, we provide sufficient conditions to ensure the existence of a common error bound for each of the systems obtained from the original one when the given data undergoes a small perturbation. Moreover, we also give some dual characterizations on the existence of error bounds for systems of conic inequalities.

Error Bounds for Some Convex Functions and Distance Composite Functions

Kung Fu Ng and Wei Hong Yang

SIAM J. Optim. 15, pp. 1042-1056 (15 pages)

Online Publication Date: July 28, 2006

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In this paper we study error bounds for function $\varphi(x)={\rm dist}(Ax+b,K)$, where $A$ is a continuous linear operator from a Banach space $X$ into another Banach space $Y$, $b\in Y$, and $K\subset Y$ is a closed set. Assuming that $A$ is of closed range ${\rm Im}(A)$, $\varphi$ is shown to have a Lipschitz error bound ($\Upsilon$-error bound) if and only if $\{{\rm Im}(A)+b,K\}$ is linearly regular (regular). Next we identify exactly when the convex function $g=\sqrt{\zeta}-l$ has error bounds (of various types), where $\zeta$ is a quadratic function satisfying $\zeta\geq 0$ and $l$ is an affine function on $\mathbb{R}^{n}$. As an application, we study the existence of various error boundsfor the function $\varphi:x\mapsto {\rm dist}(Ax+b,\mathcal{K})$ to supplement the results reported in an earlier paper [K. F. Ng and W. H. Yang, SIAM J. Optim., 13 (2002), pp. 24--43] for the standard second-order cone $\mathcal{K}$ in $\mathbb{R}^{n}$.

An Algorithm Model for Mixed Variable Programming

S. Lucidi, V. Piccialli, and M. Sciandrone

SIAM J. Optim. 15, pp. 1057-1084 (28 pages) | Cited 8 times

Online Publication Date: July 28, 2006

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In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed variable optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular, we define a general algorithm model for the solution of this class of problems, that draws inspiration from the approach recently proposed by Audet and Dennis [SIAM J. Optim., 11 (2001), pp. 573--594], and is based on the strategy of combining in a suitable way a local search with respect to the continuous variables and a local search with respect to the discrete variables. We prove global convergence of the algorithm model without specifying the local continuous search, but only identifying some reasonable requirements. Moreover, we define a particular derivative-free algorithm for solving mixed variable programming problems where the continuous variables are linearly constrained and derivative information is not available. Finally, we report numerical results obtained by the proposed algorithm in solving a real optimal design problem. These results show the effectiveness of the approach.

Extreme Points of Sets of Randomized Strategies in Constrained Optimization and Control Problems

Juan González-Hernández and Onésimo Hernández-Lerma

SIAM J. Optim. 15, pp. 1085-1104 (20 pages) | Cited 5 times

Online Publication Date: July 28, 2006

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This paper concerns the existence and characterization of optimal randomized strategies for some constrained optimization and control problems. We first present a characterization of the extreme points of a set of randomized strategies that satisfy n moment-like constraints. Conditions are given under which those extreme points are randomizations of at most n+1 deterministic strategies. This result is then applied to obtain the existence and characterization of optimal strategies for a class of deterministic, allocation-like, optimization problems and their Young relaxations. Similar results are obtained for constrained Markov control processes in Borel spaces.

A Predictor-Corrector Algorithm for Linear Optimization Based on a Specific Self-Regular Proximity Function

Jiming Peng, Tamás Terlaky, and Yunbin Zhao

SIAM J. Optim. 15, pp. 1105-1127 (23 pages) | Cited 11 times

Online Publication Date: July 28, 2006

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It is well known that the so-called first-order predictor-corrector methods working in a large neighborhood of the central path are among the most efficient interior-point methods (IPMs) for linear optimization (LO) problems. However, the best known iteration complexity of this type of method is $O(n \log\frac{(x^0)^Ts^0}{\varepsilon})$. It is of interest to investigate whether the complexity of first-order predictor-corrector type methods can be further improved. In this paper, based on a specific self-regular proximity function, we define a new large neighborhood of the central path. In particular, we show that the new neighborhood matches the standard large neighborhood that is defined by the infinity norm and widely used in the IPM literature. A new first-order predictor-corrector method for LO that uses a search direction induced by self-regularity in corrector steps is proposed. We prove that our predictor-corrector algorithm, working in a large neighborhood, has an $O(\sqrt{n}\log n \log\frac{(x^0)^Ts^0}{\varepsilon})$ iteration bound. Local superlinear convergence of the algorithm is also established.

On Saddle Points of Augmented Lagrangians for Constrained Nonconvex Optimization

X. L. Sun, D. Li, and K. I. M. McKinnon

SIAM J. Optim. 15, pp. 1128-1146 (19 pages) | Cited 17 times

Online Publication Date: July 28, 2006

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We present in this paper new results on the existence of saddle points of augmented Lagrangian functions for constrained nonconvex optimization. Four classes of augmented Lagrangian functions are considered: the essentially quadratic augmented Lagrangian, the exponential-type augmented Lagrangian, the modified barrier augmented Lagrangian, and the penalized exponential-type augmented Lagrangian. We first show that under second-order sufficiency conditions, all these augmented Lagrangian functions possess local saddle points. We then prove that global saddle points of these augmented Lagrangian functions exist under certain mild additional conditions. The results obtained in this paper provide a theoretical foundation for the use of augmented Lagrangians in constrained global optimization. Our findings also give new insights to the role played by augmented Lagrangians in local duality theory of constrained nonconvex optimization.

A Note on the Local Convergence of a Predictor-Corrector Interior-Point Algorithm for the Semidefinite Linear Complementarity Problem Based on the Alizadeh--Haeberly--Overton Search Direction

Zhaosong Lu and Renato D. C. Monteiro

SIAM J. Optim. 15, pp. 1147-1154 (8 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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This note points out an error in the local quadratic convergence proof of the predictor-corrector interior-point algorithm for solving the semidefinite linear complementarity problem based on the Alizadeh--Haeberly--Overton search direction presented in [M. Kojima, M. Shida, and S. Shindoh, SIAM J. Optim., 9 (1999), pp. 444--465]. Their algorithm is slightly modified and the local quadratic convergence of the resulting method is established.

On the Stability of the Extreme Point Set in Linear Optimization

Miguel A. Goberna, Mercedes Larriqueta, and Virginia Vera de Serio

SIAM J. Optim. 15, pp. 1155-1169 (15 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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This paper analyzes the stability properties of the set of extreme points of a closed convex set described by means of a given linear inequality system $\sigma$. We assume that all the coefficients of $\sigma$ can be arbitrarily perturbed maintaining the (possibly infinite) index set as well as the (finite) dimension of the space of variables, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. The paper characterizes the nonemptiness of the extreme point set under sufficiently small perturbations and the Berge lower semicontinuity of the extreme point set mapping at $\sigma$. It also gives necessary as well as sufficient conditions for the closedness and for the Berge upper semicontinuity at $\sigma$ which provide characterizations of these properties for finite systems.

Grover's Quantum Algorithm Applied to Global Optimization

W. P. Baritompa, D. W. Bulger, and G. R. Wood

SIAM J. Optim. 15, pp. 1170-1184 (15 pages) | Cited 7 times

Online Publication Date: July 28, 2006

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Grover's quantum computational search procedure can provide the basis for implementing adaptive global optimization algorithms. A brief overview of the procedure is given and a framework called Grover adaptive search is set up. A method of Dürr and Hoyer and one introduced by the authors fit into this framework and are compared.

Flows over Time with Load-Dependent Transit Times

Ekkehard Köhler and Martin Skutella

SIAM J. Optim. 15, pp. 1185-1202 (18 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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More than forty years ago, Ford and Fulkerson studied maximum s-t-flows over time (also called "dynamic" flows) in networks with fixed transit times on the arcs and a fixed time horizon. Here, flow on arcs may change over time and transit times specify the amount of time it takes for flow to travel through a particular arc. Ford and Fulkerson proved that there always exists an optimal solution which sends flow on certain s-t-paths at a constant rate as long as there is enough time left for the flow along a path to arrive at the sink; a flow over time featuring this simple structure is called "temporally repeated."
Although this result does not hold for the more general and also more realistic setting where transit times depend on the current flow situation, we show that there always exists a provably good temporally repeated solution. Moreover, such a solution can be determined very efficiently by only one minimum convex cost flow computation. Our results rest upon a new model of flow-dependent transit times. It is based on two assumption on the pace of flow on a particular arc. First, the pace of flow on an arc is assumed to be uniform for all flow units on an arc for each point in time. Second, this uniform pace is for each moment determined by the actual amount of flow on this arc. Finally, we show that the resulting flow-over-time problem is strongly NP-hard and cannot be approximated with arbitrary precision in polynomial time, unless P=NP.

On Using the Elastic Mode in Nonlinear Programming Approaches to Mathematical Programs with Complementarity Constraints

Mihai Anitescu

SIAM J. Optim. 15, pp. 1203-1236 (34 pages) | Cited 25 times

Online Publication Date: July 28, 2006

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We investigate the possibility of solving mathematical programs with complementarity constraints (MPCCs) using algorithms and procedures of smooth nonlinear programming. Although MPCCs do not satisfy a constraint qualification, we establish sufficient conditions for their Lagrange multiplier set to be nonempty. MPCCs that have nonempty Lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter. In this context, the elastic mode transforms MPCC into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem, which in turn makes it approachable by sequential quadratic programming (SQP) algorithms. One such algorithm is shown to achieve local linear convergence once the problem is relaxed. Under stronger conditions, we also prove superlinear convergence to the solution of an MPCC using an adaptive elastic mode approach for an SQP algorithm recently analyzed in an MPCC context in [R. Fletcher, S. Leyffer, S. Sholtes, and D. Ralph, Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints, Tech. report NA 210, University of Dundee, Dundee, UK, 2002]. Our assumptions are more general since we do not use a critical assumption from that reference. In addition, we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen.

Tail Decay and Moment Estimates of a Condition Number for Random Linear Conic Systems

Dennis Cheung, Felipe Cucker, and Raphael Hauser

SIAM J. Optim. 15, pp. 1237-1261 (25 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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In this paper we study the distribution of $\mathscr C(A)$ and $\log\mathscr C(A)$, where $\mathscr C(A)$ is a condition number for the linear conic system $Ax\leq 0$, $x\neq 0$, with $A\in\Bbb R^{n\times m}$. For Gaussian matrices $A$ we develop both upper and lower bounds on the decay rates of the distribution tails of $\mathscr C(A)$, showing that ${\bf P}\left[\mathscr C(A)\geq t\right]\sim c/t$ for large $t$, where $c$ is a factor that depends only on the problem dimensions $(m,n)$. Using these bounds, we derive moment estimates for $\mathscr C(A)$ and $\log\mathscr C(A)$ and prove various limit theorems for the cases where $m$ and/or $n$ are large. Combined with condition number based complexity analyses, our results yield tail information on the distribution of running times for interior-point or relaxation methods designed to solve the feasibility problem $Ax\leq 0$, $x\neq 0$.
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