arXiv subjects

Michal Kočvara

Publications and source records attributed to Michal Kočvara.

6 recordsLinked to original sources

arXiv2026

On the role of semismoothness in the implicit programming approach to selected nonsmooth optimization problems

The paper deals with the implicit programming approach to a class of Mathematical Programs with Equilibrium Constraints (MPECs) and bilevel programs in the case when the corresponding reduced problems are solved using a bundle method of nonsmooth optimization. The obtained results allow us to supply the bundle algorithm with suitable, easily computable ``pseudosubgradients'', ensuring convergence to points satisfying a stationary condition. Both the theory and computational implementation heavily rely on the notion of SCD (subspace containing derivatives) mappings and the associated calculus. The approach is validated via a complex MPEC with equilibrium governed by a variational inequality of the 2nd kind, by a shape optimization problem with a nonsmooth objective and by an academic bilevel program with a nonsmooth upper-level objective.

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arXiv2025

Global weight optimization of frame structures under free-vibration eigenvalue constraints

Topology optimization of frame structures under free-vibration eigenvalue constraints constitutes a challenging nonconvex polynomial optimization problem with disconnected feasible sets. In this article, we first formulate it as a polynomial semidefinite programming problem (SDP) of minimizing a linear function over a basic semi-algebraic feasible set. We then propose to solve this problem by Lasserre hierarchy of linear semidefinite relaxations providing a sequence of increasing lower bounds. To obtain also a sequence of upper bounds and thus conditions on global $\varepsilon$-optimality, we provide a bilevel reformulation that exhibits a special structure: The lower level is quasiconvex univariate and it has a non-empty interior if the constraints of the upper-level problem are satisfied. After deriving the conditions for the solvability of the lower-level problem, we thus provide a way to construct feasible points to the original SDP. Using such a feasible point, we modify the original nonlinear SDP to satisfy the conditions for the deployment of the Lasserre hierarchy. Solving arbitrary degree relaxation of the hierarchy, we prove that scaled first-order moments associated with the problem variables satisfy feasibility conditions for the lower-level problem and thus provide guaranteed upper and lower bounds on the objective function. Using these bounds, we develop a simple sufficient condition for global $\varepsilon$-optimality and prove that the optimality gap $\varepsilon$ converges to zero if the set of global minimizers is convex. Finally, we illustrate these results with four representative problems for which the hierarchy converges in at most five relaxation degrees

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arXiv2025

Moment-SOS hierarchies for arrow-type polynomial matrix inequalities with applications to structural optimization

The Arrow Decomposition (AD) technique, initially introduced in [Mathematical Programming 190(1-2) (2021), pp 105-134], demonstrated superior scalability over the classical chordal decomposition in the context of Linear Matrix Inequalities (LMIs) if the matrix in question satisfied suitable assumptions. The primary objective of this paper is to extend the AD method to address Polynomial Optimization Problems (POPs) involving large-scale Polynomial Matrix Inequalities (PMIs), with the solution framework relying on moment-sum of square (mSOS) hierarchies. As a first step, we revisit the LMI case and weaken the conditions necessary for the key AD theorem presented in [Mathematical Programming 190(1-2) (2021), pp 105-134]. This modification allows the method to be applied to a broader range of problems. Next, we propose a practical procedure that reduces the number of additional variables, drawing on physical interpretations often found in structural optimization applications. For the PMI case, we explore two distinct approaches to combine the AD technique with mSOS hierarchies. One approach involves applying AD to the original POP before implementing the mSOS relaxation. The other approach applies AD directly to the mSOS relaxations of the POP. We establish convergence guarantees for both approaches and prove that theoretical properties extend to the polynomial case. Finally, we illustrate the significant computational advantages offered by the application of AD, particularly in the context of structural optimization problems.

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arXiv2025

Term-sparse polynomial optimization for the design of frame structures

This work investigates an efficient solution to two fundamental problems in topology optimization of frame structures. The first one involves minimizing structural compliance under linear-elastic equilibrium and weight constraint, while the second one minimizes the weight under compliance constraints. These problems are non-convex and generally challenging to solve globally, with the non-convexity concentrated in a polynomial matrix inequality. We solve these problems using the moment-sum-of-squares (mSOS) hierarchy and improve the scalability by enhancing (mSOS) with the Term Sparsity Pattern (TSP) technique. Additionally, we exploit the unique polynomial structure of our problems by adopting a reduced monomial basis containing only non-mixed terms. These modifications significantly enhance computational efficiency. Extensive numerical experiments demonstrate that our approach achieves global solutions for instances twice as large as those previously solved while substantially accelerating the solution process.

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arXiv2023

Global weight optimization of frame structures with polynomial programming

Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic structure and adopt the Lasserre hierarchy of relaxations to compute the global minimizers. While this hierarchy generates a natural sequence of lower bounds, we show, under mild assumptions, how to project the relaxed solutions onto the feasible set of the original problem and thus construct feasible upper bounds. Based on these bounds, we develop a simple sufficient condition of global $\varepsilon$-optimality. Finally, we prove that the optimality gap converges to zero in the limit if the set of global minimizers is convex. We demonstrate these results by means of two academic illustrations.

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arXiv2013

PENLAB: A MATLAB solver for nonlinear semidefinite optimization

PENLAB is an open source software package for nonlinear optimization, linear and nonlinear semidefinite optimization and any combination of these. It is written entirely in MATLAB. PENLAB is a young brother of our code PENNON \cite{pennon} and of a new implementation from NAG \cite{naglib}: it can solve the same classes of problems and uses the same algorithm. Unlike PENNON, PENLAB is open source and allows the user not only to solve problems but to modify various parts of the algorithm. As such, PENLAB is particularly suitable for teaching and research purposes and for testing new algorithmic ideas. In this article, after a brief presentation of the underlying algorithm, we focus on practical use of the solver, both for general problem classes and for specific practical problems.

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