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SIAM J. on Optimization

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2003

Volume 13, Issue 4, pp. 947-1244


Characterizing Set Containments Involving Infinite Convex Constraints and Reverse-Convex Constraints

V. Jeyakumar

SIAM J. Optim. 13, pp. 947-959 (13 pages) | Cited 28 times

Online Publication Date: July 28, 2006

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Dual characterizations of the containment of a closed convex set, defined by infinite convex constraints, in an arbitrary polyhedral set, in a reverse-convex set, defined by convex constraints, and in another convex set, defined by finite convex constraints, are given. A special case of these dual characterizations has played a key role in generating knowledge-based support vector machine classifiers which are powerful tools in data classification and mining. The conditions in these dual characterizations reduce to simple nonasymptotic conditions under Slater's constraint qualification.

Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems

Xin Chen, Houduo Qi, and Paul Tseng

SIAM J. Optim. 13, pp. 960-985 (26 pages) | Cited 21 times

Online Publication Date: July 28, 2006

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For any function f from $\mathbb R$ to $\mathbb R$, one can define a corresponding function on the space of n × n (block-diagonal) real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as ($\rho$-order) semismoothness. Our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices. We also apply our results to the semidefinite complementarity problem, addressing some basic issues in the analysis of smoothing/semismooth Newton methods for solving this problem.

Epiconvergence of Convexly Composite Functions in Banach Spaces

Christophe Combari and Lionel Thibault

SIAM J. Optim. 13, pp. 986-1003 (18 pages)

Online Publication Date: July 28, 2006

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This paper gives conditions ensuring the epiconvergence or $\Gamma$-convergence of sequences of convexly composite functions via the study of Painlevé--Kuratowski convergence of inverse images of sets. The convergence of multipliers of optimization problems associated with convexly composite functions is also established.

On the Primal-Dual Geometry of Level Sets in Linear and Conic Optimization

Robert M. Freund

SIAM J. Optim. 13, pp. 1004-1013 (10 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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For a conic optimization problem $$ \begin{array}{lclr} P: & {\rm minimize}_x & c^{T}x \\ & \mbox{s.t. } & Ax=b,\\ & & x \in C \\ \end{array} $$ \noindent and its dual $$ \begin{array}{lclr} D: & {\rm supremum}_{y,s} & b^{T}y\\ & \mbox{ s.t. } & A^Ty+s=c,\\ & & s \in C^* ,\\ \end{array} $$ we present a geometric relationship between the primal objective function level sets and the dual objective function level sets, which shows that the maximum norms of the primal objective function level sets are nearly inversely proportional to the maximum inscribed radii of the dual objective function level sets.

Computational Experience with Stable Set Relaxations

Gerald Gruber and Franz Rendl

SIAM J. Optim. 13, pp. 1014-1028 (15 pages) | Cited 8 times

Online Publication Date: July 28, 2006

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We investigate relaxations for the maximum stable set problem based on the Lovász number $\vartheta(G)$ as an initial upper bound. We strengthen this relaxation by adding two classes of cutting planes, odd circuit and triangle inequalities. We present computational results using this tighter model on many classes of graphs.

The Analytic Center Cutting Plane Method with Semidefinite Cuts

Mohammad R. Oskoorouchi and Jean-Louis Goffin

SIAM J. Optim. 13, pp. 1029-1053 (25 pages) | Cited 7 times

Online Publication Date: July 28, 2006

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We analyze an analytic center cutting plane algorithm for convex feasibility problems with semidefinite cuts. The problem of interest seeks a feasible point in a bounded convex set, which contains a full-dimensional ball with $\varepsilon$ $(<1)$ radius and is contained in a compact convex set described by matrix inequalities, known as the set of localization. At each iteration, an approximate analytic center of the set of localization is computed. If this point is not in the solution set, an oracle is called to return a p-dimensional semidefinite cut. The set of localization is then updated by adding the semidefinite cut through the center. We prove that the analytic center is recovered after adding a pk-dimensional semidefinite cut in O(pk log(pk+1)) damped Newton's iteration and that the algorithmstops with a point in the solution set when the dimension of the accumulated block diagonal cut matrix reaches the bound of $O^*(\frac{p^2m^3}{{\mu^2\varepsilon}^2})$, where p is the maximum dimension of the cut matrices and $\mu>0$ is a condition number of the field of cuts.

A Variant of the Vavasis--Ye Layered-Step Interior-Point Algorithm for Linear Programming

Renato D. C. Monteiro and Takashi Tsuchiya

SIAM J. Optim. 13, pp. 1054-1079 (26 pages) | Cited 8 times

Online Publication Date: July 28, 2006

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In this paper we present a variant of Vavasis and Ye's layered-step path-following primal-dual interior-point algorithm for linear programming. Our algorithm is a predictor--corrector-type algorithm which uses from time to time the layered least squares (LLS) direction in place of the affine scaling (AS) direction. It has the same iteration-complexity bound of Vavasis and Ye's algorithm, namely ${\cal O}(n^{3.5}\log({\bar\chi_A}+n))$, where n is the number of nonnegative variables and ${\bar\chi_A}$ is a certain condition number associated with the constraint matrix A. Vavasis and Ye's algorithm requires explicit knowledge of ${\bar\chi_A}$ (which is very hard to compute or even estimate) in order to compute the layers for the LLS direction. In contrast, our algorithm uses the AS direction at the current iterate to determine the layers for the LLS direction, and hence does not require the knowledge of ${\bar\chi_A}$. A variant with similar properties and with the same complexity has been developed by Megiddo, Mizuno, and Tsuchiya [Math. Programming, 82 (1998), pp. 339--355]. However, their algorithm needs to compute n LLS directions on every iteration, while ours computes at most one LLS direction on any given iteration.

Inexact Variants of the Proximal Point Algorithm without Monotonicity

A. N. Iusem, T. Pennanen., and B. F. Svaiter

SIAM J. Optim. 13, pp. 1080-1097 (18 pages) | Cited 14 times

Online Publication Date: July 28, 2006

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This paper studies convergence properties of inexact variants of the proximal point algorithm when applied to a certain class of nonmonotone mappings. The presented algorithms allow for constant relative errors, in the line of the recently proposed hybrid proximal-extragradient algorithm. The main convergence result extends a recent work of the second author, where exact solutions for the proximal subproblems were required. We also show that the linear convergence property is preserved in the case when the inverse of the operator is locally Lipschitz continuous near the origin. As an application, we give a convergence analysis for an inexact version of the proximal method of multipliers for a rather general family of problems which includes variational inequalities and constrained optimization problems.

A Sequential Quadratically Constrained Quadratic Programming Method for Differentiable Convex Minimization

Masao Fukushima, Zhi-Quan Luo, and Paul Tseng

SIAM J. Optim. 13, pp. 1098-1119 (22 pages) | Cited 16 times

Online Publication Date: July 28, 2006

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This paper presents a sequential quadratically constrained quadratic programming (SQCQP) method for solving smooth convex programs. The SQCQP method solves at each iteration a subproblem that involves convex quadratic inequality constraints as well as a convex quadratic objective function. Such a quadratically constrained quadratic programming problem can be formulated as a second-order cone program, which can be solved efficiently by using interior point methods. We consider the following three fundamental issues on the SQCQP method: the feasibility of subproblems, the global convergence, and the quadratic rate of convergence. In particular, we show that the Maratos effect is avoided without any modification to the search direction, even though we use an ordinary $\ell_1$ exact penalty function as the line search merit function.

Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Simple Bounds and Equality Constraints

Z. Dostál, A. Friedlander, and S. A. Santos

SIAM J. Optim. 13, pp. 1120-1140 (21 pages) | Cited 15 times

Online Publication Date: July 28, 2006

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In this paper we discuss a specialization of the augmented Lagrangian-type algorithm of Conn, Gould, and Toint to the solution of strictly convex quadratic programming problems with simple bounds and equality constraints. The new feature of the presented algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm which yields a rate of convergence that does not have any term that accounts for inexact solution of auxiliary problems. Moreover, boundedness of the penalty parameter is achieved for the precision control used. Numerical experiments illustrate the efficiency of the presented algorithm and encourage its usage.

A New Exact Penalty Function

Waltraud Huyer and Arnold Neumaier

SIAM J. Optim. 13, pp. 1141-1158 (18 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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For constrained smooth or nonsmooth optimization problems, new continuously differentiable penalty functions are derived. They are proved exact in the sense that under some nondegeneracy assumption, local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function. This is achieved by augmenting the dimension of the program by a variable that controls both the weight of the penalty terms and the regularization of the nonsmooth terms.

Iterating Bregman Retractions

Heinz H. Bauschke and Patrick L. Combettes

SIAM J. Optim. 13, pp. 1159-1173 (15 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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The notion of a Bregman retraction of a closed convex set in Euclidean space is introduced. Bregman retractions include backward Bregman projections and forward Bregman projections, as well as their convex combinations, and are thus quite flexible. The main result on iterating Bregman retractionsunifies several convergence results on projection methods for solving convex feasibility problems. It is also used to construct new sequential and parallel algorithms.

Primal-Dual Gradient Structured Functions: Second-Order Results; Links to Epi-Derivatives and Partly Smooth Functions

Robert Mifflin and Claudia Sagastizábal

SIAM J. Optim. 13, pp. 1174-1194 (21 pages) | Cited 13 times

Online Publication Date: July 28, 2006

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We give second-order expansions for quite general nonsmooth functions from the $\cal{V}\cal{U}$-space decomposition point of view. The results depend on primal-dual gradient structure, which we relate to general concepts of second-order epi-derivatives and partly smooth functions. Expressions for the associated second-order objects are given in terms of $\cal{U}$-subspace Hessians.

A Globally and Locally Superlinearly Convergent Non--Interior-Point Algorithm for P0 LCPs

Yun-Bin Zhao and Duan Li

SIAM J. Optim. 13, pp. 1195-1221 (27 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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Based on the concept of the regularized central path, a new non--interior-point path-following algorithm is proposed for solving the P0 linear complementarity problem (P0 LCP). The condition ensuring the global convergence of the algorithm for P0 LCPs is weaker than most conditions previously used in the literature. This condition can be satisfied even when the strict feasibility condition, which has often been assumed in most existing non--interior-point algorithms, fails to hold. When the algorithm is applied to P* and monotone LCPs, the global convergence of this method requires no assumption other than the solvability of the problem. The local superlinear convergence of the algorithm can be achieved under a nondegeneracy assumption. The effectiveness of the algorithm is demonstrated by our numerical experiments.

A Feasible Sequential Linear Equation Method for Inequality Constrained Optimization

Yu-Fei Yang, Dong-Hui Li, and Liqun Qi

SIAM J. Optim. 13, pp. 1222-1244 (23 pages) | Cited 7 times

Online Publication Date: July 28, 2006

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In this paper, by means of the concept of the working set, which is an estimate of the active set, we propose a feasible sequential linear equation algorithm for solving inequality constrained optimization problems. At each iteration of the proposed algorithm, we first solve one system of linear equations with a coefficient matrix of size m × m (where m is the number of constraints) to compute the working set; we then solve a subproblem which consists of four reduced systems of linear equations with a common coefficient matrix. Unlike existing QP-free algorithms, the subproblem is concerned with only the constraints corresponding to the working set. The constraints not in the working set are neglected. Consequently, the dimension of each subproblem is not of full dimension. Without assuming the isolatedness of the stationary points, we prove that every accumulation point of the sequence generated by the proposed algorithm is a KKT point of the problem. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. In other words, after finitely many steps, only those constraints which are active at the solution will be involved in the subproblem. Under some additional conditions, we show that the convergence rate is two-step superlinear or even Q-superlinear. We also report some preliminary numerical experiments to show that the proposed algorithm is practicable and effective for the test problems.
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