Hans Frenk

High Performance Optimization

List of Figures. List of Tables. Preface. Contributing Authors. Part I: Theory and Algorithms of Semidefinite Programming J.F. Sturm. 1. Introduction. 2. Duality. 3. Polynomiality of Path-following Methods. 4. Self-Dual Embedding Technique. 5. Properties of the Central Path. 6. Superlinear Convergence. 7. Central Region Methods. Part II: Linear, Quadratic, Semidefinite Programming and Beyond. 8. An Implementation of the Homogeneous Algorithm E.D. Andersen, K.D. Andersen. 9. A Simplified Correctness Proof for Interior Point Algorithm S.A. Vavasis, Y. Ye. 10. New Analysis of Newton Methods for LCP J. Peng, et al. 11. Numerical Evaluation of SDPA K. Fujisawa, et al. 12. Robust Modeling of Multi-Stage Portfolio Problems A. Ben-Tal, et al. 13. An Interior Point SQP Parallel B&B Method. 14. Solving Linear Ordering Problems J.E. Mitchell, B. Borchers. 15. Finite Element Methods for Solving Parabolic Inverse Problems Y.L. Keung, J. Zou. 16. Error Bounds For Quadratic Systems Z.-Q. Luo, J.F. Sturm. 17. Squared Functional Systems and Optimization Problems Y. Nesterov. 18. Interior Point Methods: Current Status and Future Directions R.M. Freund, S. Mizuno. Index.

Improved Algorithms for Machine Allocation in Manufacturing Systems

In this paper we present two algorithms for a machine allocation problem occurring in manufacturing systems. For the two algorithms presented we prove worst-case performance ratios of 2 and 312, respectively. The machlne allocat~on problem we consider is a general convex resource allocation problem, which makes the algorithms applicable to a varlety of resource allocation problems. Numerical results are presented for two real-life manufacturing systems.

How to Determine Maintenance Frequencies for Multi-Component Systems? A General Approach

A maintenance activity carried out on a technical system often involves a system-dependent set-up cost that is the same for all maintenance activities carried out on that system. Grouping activities thus saves costs since execution of a group of activities requires only one set-up. By now, there are several multi-component maintenance models available in the literature, but most of them suffer from intractability when the number of components grows, unless a special structure is assumed. An approach that can handle many components was introduced in the literature by Goyal et al. However, this approach requires a specific deterioration structure for components. Moreover, the authors present an algorithm that is not optimal and there is no information of how good the obtained solutions are. In this paper, we present an approach that solves the model of Goyal et al. to optimality. Furthermore, we extend the approach to deal with more general maintenance models like minimal repair and inspection that can be solved to optimality as well. Even block replacement can be incorporated, in which case our approach is a good heuristic.

A Deterministic Inventory/Production Model with General Inventory Cost Rate Function and Concave Production Costs

We present a thorough analysis of the economic order quantity model with shortages under a general inventory cost rate function and concave production costs. By using some standard results from convex analysis, we show that the model exhibits a composite concave-convex structure. Consequently, an effective solution procedure, particularly useful for an approximation scheme, is proposed. A computational study is appended to illustrate the performance of the proposed solution procedure.

Insurers' profits in the third-party liability insurance

In this note we derive the expected total discounted profit of an insurer due to a single policy holder within a third-party liability insurance. We consider both a policy holder claiming optimally and non-optimally.

International workshop on smooth and nonsmooth optimization (Rotterdam, July 12–13, 2001)

The Workshop was held right after the EURO2001 meeting. It was organized by the EURO Working Group on Continuous Optimization (EUROPT). The workshop is one of the workshops in conference series organized by EUROPT, the preceding one, Interior Point Methods 2000 was held directly before the EURO2000 conference in Budapest, and in 2003 the Advances in Continuous Optimization workshop is held preceding the EURO2003 meeting in Istanbul. The executing organizing committee of the International Workshop on Smooth and Nonsmooth Optimization was formed by: Jan Brinkhuis, Tibor Ill es, Hans Frenk, Georg Still and Gerhard Weber. The workshop aimed to bring together researchers from smooth and nonsmooth optimization and from related fields of discrete optimization, operations research, economy and technology. It provided a forum for the exchange of recent scientific developments and for the discussion of new trends. The scope of the conference included all aspects of smooth, nonsmooth and discrete optimization from fundamental research to numerical methods and applications. The workshop was trade-marked by the six speakers: P. Gritzmann (Technical University of M€ unchen), H.Th. Jongen (University of Aachen), A. Kruger (Institute of Labour and Social Relations, Minsk), B. Polyak (Institute of Control Science, Moscow), V. Protassov (Moscow State University), and K. Roos (Technical University Delft). More than 40 participants contributed 14 additional presentations.

Polynomiality of Path-Following Methods

Modern solvers for linear and quadratic programming often incorporate some variant of the primal-dual path-following method, which is a class of interior point methods. This method turns out to be suited for solving semidefinite programming problems as well. In this chapter, we propose a framework for analyzing primal-dual path-following algorithms for semidefinite programming. Our framework is a direct extension of the v-space approach, which was developed by Kojima et al. [63] in the context of complementarity problems.

Properties of the Central Path

In this chapter, we take a careful look at the behavior of the central path, when it approaches the optimal solution set of a semidefinite program. We will demonstrate that the primal-dual central path converges to the analytic center of the optimal solution set. Moreover, the distance to this analytic center from any point on the central path is shown to converge at the same R-rate as the duality gap. This result can be interpreted as an error-bound for solutions on the central path, with respect to the optimal solution set. Underlying the analysis is an assumption that a strictly complementary solution pair for the semidefinite program exists.

Central Region Method

In this chapter, we discuss a modification of the standard path-following scheme that tends to speed up the global convergence. This modification, the central region method, generates iterates that do not really trace the central path, or at least not closely. In this way, it has a relatively large freedom of movement, and consequently the ability to take long steps. This makes it interesting to consider more sophisticated search directions. We propose a search direction that is built up in three phases, viz. 1. Initial centering, 2. Predictor, 3. Second order centrality corrector.

Self-Dual Embedding Technique

In Chapter 3, we have analyzed the iteration complexity of finding an ϵ-optimal solution, if an interior, sufficiently centered pair of primal and dual solutions is known beforehand. We will see in this chapter how we can adapt the algorithms of Chapter 3 to solve semidefinite programming problems without any pre-knowledge. To this end, we use the self-dual embedding technique. This technique will also be used to tackle semidefinite programming problems that may be unbounded, unsolvable, or infeasible.