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SIAM J. on Optimization

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2006

Volume 16, Issue 4, pp. 939-1241


Discrete Transforms, Semidefinite Programming, and Sum-of-Squares Representations of Nonnegative Polynomials

Tae Roh and Lieven Vandenberghe

SIAM J. Optim. 16, pp. 939-964 (26 pages) | Cited 11 times

Online Publication Date: July 28, 2006

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We present a new semidefinite programming formulation of sum-of-squares representations of nonnegative polynomials, cosine polynomials, and trigonometric polynomials of one variable. The parametrization is based on discrete transforms (specifically, the discrete Fourier, cosine, and polynomial transforms) and has a simple structure that can be exploited by straightforward modifications of standard interior-point algorithms.

Incorporating Condition Measures in the Context of Combinatorial Optimization

Jorge R. Vera and Iván Derpich

SIAM J. Optim. 16, pp. 965-985 (21 pages)

Online Publication Date: July 28, 2006

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Integer programming algorithms have some kind of exponential complexity in the worst case. However, it is also observed that data instances of similar sizes might have very different practical complexity when solved by computer algorithms. This paper considers an alternative complexity analysis of some integer programming algorithms, which is based on measures of "intrinsic difficulty." The work extends to the setup of integer programming some notions of condition measures which have been developed for convex optimization. We present bounds on the so-called lattice width of polyhedra and address the impact on the complexity of integer programming algorithms like Lenstra's algorithm as well as branch and bound algorithms. The condition measures introduced here reflect shape and spatial orientation factors which are not fully captured by the traditional combinatorial analysis.

Fast Algorithms for Projection on an Ellipsoid

Yu-Hong Dai

SIAM J. Optim. 16, pp. 986-1006 (21 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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Several fast algorithms are proposed for the problem of projecting a point onto a general ellipsoid. To avoid the direct estimation of the spectral radius in the Lin--Han algorithm, we provide the maximal $2$-dimensional inside ball algorithm and the sequential $2$-dimensional projection algorithm. However, we find that the solution procedure of the former algorithm may tend to generate some $2$-dimensional reduced ellipsoids, and the latter algorithm may produce zigzags. Therefore we investigate the hybrid use of the two algorithms. Our numerical experiments show that all the algorithms, even the hybrid algorithms, are suitable for large-scale problems and much faster than the Lin--Han algorithm. Linear convergence of the algorithms is established. Possible extensions of the algorithms are also discussed.

A Proximal Bundle Method with Approximate Subgradient Linearizations

Krzysztof C. Kiwiel

SIAM J. Optim. 16, pp. 1007-1023 (17 pages) | Cited 14 times

Online Publication Date: July 28, 2006

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We give a proximal bundle method for minimizing a convex function $f$ over a closed convex set. It only requires evaluating $f$ and its subgradients with an accuracy $\epsilon>0$, which is fixed but possibly unknown. It asymptotically finds points that are $\epsilon$-optimal. When applied to Lagrangian relaxation, it allows for $\epsilon$-accurate solutions of Lagrangian subproblems and finds $\epsilon$-optimal solutions of convex programs.

Stochastic Order Relations and Lattices of Probability Measures

Alfred Müller and Marco Scarsini

SIAM J. Optim. 16, pp. 1024-1043 (20 pages) | Cited 1 time

Online Publication Date: July 28, 2006

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We study various partially ordered spaces of probability measures and we determine which of them are lattices. This has important consequences for optimization problems with stochastic dominance constraints. In particular we show that the space of probability measures on $\mathbb{R}$ is a lattice under most of the known partial orders, whereas the space of probability measures on $\mathbb{R}^d$ typically is not. Nevertheless, some subsets of this space, defined by imposing strong conditions on the dependence structure of the measures, are lattices.

Critical Value Functions have Finite Modulus of Concavity

Harald Günzel, Francisco Guerra Vazquez, and Hubertus Th. Jongen

SIAM J. Optim. 16, pp. 1044-1053 (10 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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We consider a smooth finite dimensional parametric optimization problem ${\cal P}(y)$ with objective function $f(x,y)$. Here, $x$ and $y$ denote the state variable and the parameter, respectively. In the case that $\overline{x}$ is a strongly stable Karush--Kuhn--Tucker point for ${\cal P}(\overline{y})$, a neighborhood of $\overline{x}$ contains a unique Karush--Kuhn--Tucker point $x(y)$ for ${\cal P}(y)$, provided that $y$ is sufficiently close to $\overline{y}$. This gives rise to the critical value function $y\mapsto\varphi(y):=f(x(y),y)$. Under the additional assumption that the Mangasarian--Fromovitz constraint qualification is satisfied at $\overline{x}$, we show that $\varphi$ has finite modulus of concavity. That means $\varphi$ becomes convex in a neighborhood of $\overline{y}$ by adding to it the function $y\mapsto (\alpha/2)\cdot\|y-\overline{y}\|^2$ for some $\alpha>0$. Moreover, we present an explicit upper bound for the $\alpha$ to be used. The latter bound turns out to be sharp for problem data in general position.

A Derivative-Free Algorithm for Linearly Constrained Finite Minimax Problems

G. Liuzzi, S. Lucidi, and M. Sciandrone

SIAM J. Optim. 16, pp. 1054-1075 (22 pages) | Cited 6 times

Online Publication Date: July 28, 2006

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In this paper we propose a new derivative-free algorithm for linearly constrained finite minimax problems. Due to the nonsmoothness of this class of problems, standard derivative-free algorithms can locate only points which satisfy weak necessary optimality conditions. In this work we define a new derivative-free algorithm which is globally convergent toward standard stationary points of the finite minimax problem. To this end, we convert the original problem into a smooth one by using a smoothing technique based on the exponential penalty function of Kort and Bertsekas. This technique depends on a smoothing parameter which controls the approximation to the finite minimax problem. The proposed method is based on a sampling of the smooth function along a suitable search direction and on a particular updating rule for the smoothing parameter that depends on the sampling stepsize. Numerical results on a set of standard minimax test problems are reported.

LMI Approximations for Cones of Positive Semidefinite Forms

Luis F. Zuluaga, Juan Vera, and Javier Peña

SIAM J. Optim. 16, pp. 1076-1091 (16 pages) | Cited 8 times

Online Publication Date: July 28, 2006

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An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices and to certain optimization problems involving random variables with some known moment information.
We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms (homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.

A New Notion of Weighted Centers for Semidefinite Programming

Chek Beng Chua

SIAM J. Optim. 16, pp. 1092-1109 (18 pages) | Cited 3 times

Online Publication Date: July 28, 2006

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The notion of weighted centers is essential in V-space interior-point algorithms for linear programming. Although there were some successes in generalizing this notion to semidefinite programming via weighted center equations, we still do not have a generalization that preserves two important properties---(1) each choice of weights uniquely determines a pair of primal-dual weighted centers, and (2) the set of all primal-dual weighted centers completely fills up the relative interior of the primal-dual feasible region. This paper presents a new notion of weighted centers for semidefinite programming that possesses both uniqueness and completeness. Furthermore, it is shown that under strict complementarity, these weighted centers converge to weighted centers of optimal faces. Finally, this convergence result is applied to homogeneous cone programming, where the central paths defined by a certain class of optimal barriers for homogeneous cones are shown to converge to analytic centers of optimal faces in the presence of strictly complementary solutions.

A Full-Newton Step O(n) Infeasible Interior-Point Algorithm for Linear Optimization

C. Roos

SIAM J. Optim. 16, pp. 1110-1136 (27 pages) | Cited 14 times

Online Publication Date: July 28, 2006

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We present a primal-dual infeasible interior-point algorithm. As usual, the algorithm decreases the duality gap and the feasibility residuals at the same rate. Assuming that an optimal solution exists, it is shown that at most $O(n)$ iterations suffice to reduce the duality gap and the residuals by the factor $1/{e}$. This implies an $O(n\log(n/\varepsilon))$ iteration bound for getting an $\varepsilon$-solution of the problem at hand, which coincides with the best known bound for infeasible interior-point algorithms. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. A special feature of the algorithm is that it uses only full-Newton steps. Two types of full-Newton steps are used, so-called feasibility steps and usual (centering) steps. Starting at strictly feasible iterates of a perturbed pair, (very) close toits central path, feasibility steps serve to generate strictly feasible iterates for the next perturbed pair. By accomplishing a few centering steps for the new perturbed pair we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The algorithm finds an optimal solution or detects infeasibility or unboundedness of the given problem.

A Truncated Projected Newton-Type Algorithm for Large-Scale Semi-infinite Programming

Qin Ni, Chen Ling, Liqun Qi, and Kok Lay Teo

SIAM J. Optim. 16, pp. 1137-1154 (18 pages) | Cited 4 times

Online Publication Date: July 28, 2006

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In this paper, a truncated projected Newton-type algorithm is presented for solving large-scale semi-infinite programming problems. This is a hybrid method of a truncated projected Newton direction and a modified projected gradient direction. The truncated projected Newton method is used to solve the constrained nonlinear system. In order to guarantee global convergence, a robust loss function is chosen as the merit function, and the projected gradient method inserted is used to decrease the merit function. This algorithm is suitable for handling large-scale problems and possesses superlinear convergence rate. The global convergence of this algorithm is proved and the convergence rate is analyzed. The detailed implementation is discussed, and some numerical tests for solving large-scale semi-infinite programming problems, with examples up to 2000 decision variables, are reported.

Combinatorial and Continuous Models for the Optimization of Traffic Flows on Networks

A. Fügenschuh, M. Herty, A. Klar, and A. Martin

SIAM J. Optim. 16, pp. 1155-1176 (22 pages) | Cited 8 times

Online Publication Date: July 28, 2006

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A hierachy of simplified models for traffic flow on networks is derived from continuous traffic flow models based on partial differential equations. The hierachy contains nonlinear and linear combinatorial models with and without dynamics. Optimization problems are treated for all models and numerical results and algorithms are compared.

A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints

M. Hintermüller and M. Hinze

SIAM J. Optim. 16, pp. 1177-1200 (24 pages) | Cited 9 times

Online Publication Date: July 28, 2006

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SQP methods for the optimal control of the instationary Navier--Stokes equations with pointwise constraints on the control are considered. Due to the presence of the constraints, the quadratic subproblems (QPs) of SQP require a more sophisticated solver when compared to the unconstrained case. In this paper, a semismooth Newton method is proposed for efficiently solving the QPs. The convergence analysis, which is performed in an appropriate function space setting, relies on the concept of slant differentiability for proving locally superlinear convergence of the QP-solver. For the analysis of the outer SQP-iteration a generalized equations approach is utilized. Sufficient conditions for guaranteeing strong regularity of the generalized equation are established which, in turn, allows one to argue a quadratic rate of convergence of the SQP-method. The paper ends with a report on numerical results supporting the theoretical findings.

Loss of Superlinear Convergence for an SQP-Type Method with Conic Constraints

Moritz Diehl, Florian Jarre, and Christoph H. Vogelbusch

SIAM J. Optim. 16, pp. 1201-1210 (10 pages)

Online Publication Date: July 28, 2006

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In this short note we consider a sequential quadratic programming (SQP)--type method with conic subproblems and compare this method with a standard SQP method in which the conic constraint is linearized at each step. For both approaches we restrict our attention to convex subproblems since these are easy to solve and guarantee a certain global descent property. Using the example of a simple nonlinear program (NLP) and its conic reformulation we show that the SQP method with conic subproblems displays a slower rate of convergence than standard SQP methods. We then explain why an SQP subproblem that is based on a better approximation of the feasible set of the NLP results in a much slower algorithm.

Polynomial Convergence of Infeasible-Interior-Point Methods over Symmetric Cones

Bharath Kumar Rangarajan

SIAM J. Optim. 16, pp. 1211-1229 (19 pages) | Cited 2 times

Online Publication Date: July 28, 2006

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We establish polynomial-time convergence of infeasible-interior-point methods for conic programs over symmetric cones using a wide neighborhood of the central path. The convergence is shown for a commutative family of search directions used in Schmieta and Alizadeh [Math. Program., 96 (2003), pp. 409-438]. Monteiro and Zhang [Math. Program., 81 (1998), pp. 281-299] introduced this family of directions when analyzing semidefinite programs. These conic programs include linear and semidefinite programs. This extends the work of Rangarajan and Todd [Tech. rep. 1388, School of OR & IE, Cornell University, Ithaca, NY, 2003], which established convergence of infeasible-interior-point methods for self-scaled conic programs using the NT direction. Our work is built on earlier analyses by Faybusovich [J. Comput. Appl. Math., 86 (1997), pp. 149-175] and Schmieta and Alizadeh [Math. Program., 96 (2003), pp. 409-438]. Of independent interest, we provide a constructive proof of Lyapunov lemma in the Jordan algebraic setting.

Strong Convergence Theorem by a Hybrid Method for Nonexpansive Mappings and Lipschitz-Continuous Monotone Mappings

Natalia Nadezhkina and Wataru Takahashi

SIAM J. Optim. 16, pp. 1230-1241 (12 pages)

Online Publication Date: July 28, 2006

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In this paper we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on two well-known methods: hybrid and extragradient. We obtain a strong convergence theorem for three sequences generated by this process. Based on this result, we also construct an iterative process for finding a common fixed point of two mappings, such that one of these mappings is nonexpansive and the other is taken from the more general class of Lipschitz pseudocontractive mappings.
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