Kees Roos

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.

The Optimal Set and Optimal Partition Approach to Linear and Quadratic Programming

In this chapter we describe the optimal set approach for sensitivity analysis for LP. We show that optimal partitions and optimal sets remain constant between two consecutive transition-points of the optimal value function. The advantage of using this approach instead of the classical approach (using optimal bases) is shown. Moreover, we present an algorithm to compute the partitions, optimal sets and the optimal value function. This is a new algorithm and uses primal and dual optimal solutions. We also extend some of the results to parametric quadratic programming, and discuss differences and resemblances with the linear programming case.

Economically Efficient Standards to Protect the Netherlands Against Flooding

In the Netherlands, flood protection is a matter of national survival. In 2008, the Second Delta Committee recommended increasing legal flood protection standards at least tenfold to compensate for population and economic growth since 1953; this recommendation would have required dike improvement investments estimated at 11.5 billion euro. Our research group was charged with developing efficient flood protection standards in a more objective way. We used cost-benefit analysis and mixed-integer nonlinear programming to demonstrate the efficiency of increasing the legal standards in three critical regions only. Monte Carlo analysis confirms the robustness of this recommendation. In 2012, the state secretary of the Ministry of Infrastructure and the Environment accepted our results as a basis for legislation. Compared to the earlier recommendation, this successful application of operations research yields both a highly significant increase in protection for these regions in which two-thirds of the benefits of the proposed improvements accrue and approximately 7.8 billion euro in cost savings. Our methods can also be used in decision making for other flood-prone areas worldwide.

On the Dimension of the Set of Rim Perturbations for Optimal Partition Invariance

Two new dimension results are presented. For linear programs, it is shown that the sum of the dimension of the optimal set and the dimension of the set of objective perturbations for which the optimal partition is invariant equals the number of variables. A decoupling principle shows that the primal and dual results are additive. The main result is then extended to convex quadratic programs, but the dimension relationships are no longer dependent only on problem size. Furthermore, although the decoupling principle does not extend completely, the dimensions are additive, as in the linear case.

Safe Dike Heights at Minimal Costs: The Nonhomogeneous Case

Dike height optimization is of major importance to the Netherlands because a large part of the country lies below sea level, and high water levels in rivers can cause floods. Recently impovements have been made on the cost-benefit model introduced by van Dantzig after the devastating flood in the Netherlands in 1953. We consider the extension of this model to nonhomogeneous dike rings, which may also be applicable to other deltas in the world. A nonhomogeneous dike ring consists of different segments with different characteristics with respect to flooding and investment costs. The individual segments can be heightened independently at different moments in time and by different amounts, making the problem considerably more complex than the homogeneous case. We show how the problem can be modeled as a mixed-integer nonlinear programming problem, and we present an iterative algorithm that can be used to solve the problem. Moreover, we consider a robust optimization approach to deal with uncertainty in the model parameters. The method has been implemented and integrated in software, which is used by the government to determine how the safety standards in the Dutch Water Act should be changed.

Polynomiality of primal-dual affine scaling algorithms for nonlinear complementarity problems

This paper provides an analysis of the polynomiality of primal-dual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu’s scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primal-dual affine scaling algorithms generates an approximate solution (given a precision ε) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial ofn, ln(1/ε) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in Jansen et al., SIAM Journal on Optimization 7 (1997) 126–140.

Self-Dual Embeddings

Most semidefinite programming algorithms found in the literature require strictly feasible starting points (X° ≻ 0, S° ≻ 0) for the primal and dual problems respectively. So-called ‘big-M’ methods (see e.g. [807]) are often employed in practice to obtain feasible starting points.

Optimal Strategies for Flood Prevention

Flood prevention policy is of major importance to the Netherlands since a large part of the country is below sea level and high water levels in rivers may also cause floods. In this paper we propose a dike height optimization model to determine economically efficient flood protection standards. We improve the model proposed by David van Dantzig [van Dantzig D (1956) Economic decision problems for flood prevention. Econometrica 24(3):276–287] after a devastating flood in the Netherlands in 1953. Our model is nonconvex, but we derive an explicit simple expression for the global optimal solution, which is periodic. We also discuss how to use this optimal investment policy to derive an efficient flood protection standard. The rather simple expression for this standard gives us much insight into how it depends on several relevant economic and climate model parameters. This approach has been applied to all dike rings in the Netherlands, and the resulting standards have been stated in the new Delta Programme 201...

On Convex Quadratic Approximation

In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of statistics and optimization. We show that convexity can be enforced in the multivariate case by using semidefinite programming techniques.

An efficient algorithm for critical circuits and finite eigenvectors in the max-plus algebra

Abstract We consider the eigenvalue problem in the max-plus algebra for matrices in {−∞∪ R } n×n but with eigenvectors in R n . The problem is relaxed to a linear optimization (LO) problem of which the dual problem is solved by finding a maximal average weight circuit in the graph of the matrix. The Floyd–Warshall procedure is used to find such a circuit. This procedure also provides an efficient algorithm for finding an eigenvector.