Michal Kočvara

Nonsmooth Approach to Optimization Problems with Equilibrium Constraints

Preface. List of Notations. List of Acronyms. Part I: Theory. 1.Introduction. 2. Auxiliary Results. 3. Algorithms of Nonsmooth Optimization. 4. Generalized Equations. 5. Stability of Solutions to Perturbed Generalized Equations. 6. Derivatives of Solutions to Perturbed Generalized Equations. 7. Optimality Conditions and a Solution Method. Part II: Applications. 8. Introduction. 9. Membrane with Obstacle. 10. Elasticity Problems with Internal Obstacles. 11. Contact Problem with Coulomb Friction. 12. Economic Applications. Appendices: A. Cookbook. B. Basic Facts on Elliptic Boundary Value problems. C. Complementarity Problems. References. Index.

PENNON: A code for convex nonlinear and semidefinite programming

We introduce a computer program PENNON for the solution of problems of convex Nonlinear and Semidefinite Programming (NLP-SDP). The algorithm used in PENNON is a generalized version of the Augmented Lagrangian method, originally introduced by Ben-Tal and Zibulevsky for convex NLP problems. We present generalization of this algorithm to convex NLP-SDP problems, as implemented in PENNON and details of its implementation. The code can also solve second-order conic programming (SOCP) problems, as well as problems with a mixture of SDP, SOCP and NLP constraints. Results of extensive numerical tests and comparison with other optimization codes are presented. The test examples show that PENNON is particularly suitable for large sparse problems.

Solving polynomial static output feedback problems with PENBMI

An algebraic formulation is proposed for the static output feedback (SOF) problem: the Hermite stability criterion is applied on the closed-loop characteristic polynomial, resulting in a non-convex bilinear matrix inequality (BMI) optimization problem for SIMO or MISO systems. As a result, the BMI problem is formulated directly in the controller parameters, without additional Lyapunov variables. The publicly available solver PENBMI 2.0 interfaced with YALMIP 3.0 is then applied to solve benchmark examples. Implementation and numerical aspects are widely discussed.

Free material optimization via mathematical programming

This paper deals with a central question of structural optimization which is formulated as the problem of finding the stiffest structure which can be made when both the distribution of material as well as the material itself can be freely varied. We consider a general multi-load formulation and include the possibility of unilateral contact. The emphasis of the presentation is on numerical procedures for this type of problem, and we show that the problems after discretization can be rewritten as mathematical programming problems of special form. We propose iterative optimization algorithms based on penalty-barrier methods and interior-point methods and show a broad range of numerical examples that demonstrates the efficiency of our approach.

Optimal Truss Design by Interior-Point Methods

This article presents a primal-dual predictor-corrector interior-point method for solving quadratically constrained convex optimization problems that arise from truss design problems. We investigate certain special features of the problem, discuss fundamental differences of interior-point methods for linearly and nonlinearly constrained problems, extend Mehrotras predictor-corrector strategy to nonlinear programs, and establish convergence of a long step method. Numerical experiments on large scale problems illustrate the surprising efficiency of the method.

A rate-independent approach to the delamination problem

We study delamination processes for elastic bodies glued together by an adhesive as an activated, rate-independent process. The adhesive is assumed to absorb a specific amount of energy during the delami-nation process. A solution is defined by energetic principles of stability and balance of stored and dissipated energies with the work of external loading, realized here through displacement on parts of the boundary. Starting from a time discretization, we construct solutions via a rigorous limiting analysis. Moreover, we provide computer simulations for some model problems using a further finite-element spatial discretization.

Optimal Design of Trusses Under a Nonconvex Global Buckling Constraint

We propose a novel formulation of a truss design problem involving a constraint on the global stability of the structure due to the linear buckling phenomenon. The optimization problem is modelled as a nonconvex semidefinite programming problem. We propose two techniques for the numerical solution of the problem and apply them to a series of numerical examples.

On the solution of large-scale SDP problems by the modified barrier method using iterative solvers

The limiting factors of second-order methods for large-scale semidefinite optimization are the storage and factorization of the Newton matrix. For a particular algorithm based on the modified barrier method, we propose to use iterative solvers instead of the routinely used direct factorization techniques. The preconditioned conjugate gradient method proves to be a viable alternative for problems with a large number of variables and modest size of the constrained matrix. We further propose to avoid explicit calculation of the Newton matrix either by an implicit scheme in the matrix–vector product or using a finite-difference formula. This leads to huge savings in memory requirements and, for certain problems, to further speed-up of the algorithm.

Multidisciplinary Free Material Optimization

We present a mathematical framework for the so-called multidisciplinary free material optimization (MDFMO) problems, a branch of structural optimization in which the full material tensor is considered as a design variable. We extend the original problem statement by a class of generic constraints depending either on the design or on the state variables. Among the examples are local stress or displacement constraints. We show the existence of optimal solutions for this generalized free material optimization (FMO) problem and discuss convergent approximation schemes based on the finite element method.

On a class of quasi-variational inequalities

In this paper we give an existence result for a class of quasi-variational inequalities. Further, we propose a nonsmooth variant of the Newton method for their numerical solution. Using the tools of sensitivity and stability theory and nonsmooth analysis, criteria are formulated ensuring the local superlinear convergence. The method is applied to the discretized contact problem with the Coulomb friction model