Michal Kocvara

University of Birmingham · School of Mathematics

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 relax-ations to compute the global minimizers. While this hierarchy...

The optimization problems with a sparsity constraint is a class of important global optimization problems. A typical type of thresholding algorithms for solving such a problem adopts the traditional full steepest descent direction or Newton-like direction as a search direction to generate an iterate on which a certain thresholding is performed. Tra...

The optimization problems with a sparsity constraint is a class of important global optimization problems. A typical type of thresholding algorithms for solving such a problem adopts the traditional full steepest descent direction or Newton-like direction as a search direction to generate an iterate on which a certain thresholding is performed. Tra...

The aim of this paper is to solve large-and-sparse linear Semidefinite Programs (SDPs) with low-rank solutions. We propose to use a preconditioned conjugate gradient method within second-order SDP algorithms and introduce a new efficient preconditioner fully utilizing the low-rank information. We demonstrate that the preconditioner is universal, in...

Discrete tomography (DT) naturally leads to a hierarchy of models of varying discretization levels. We employ multilevel optimization (MLO) to take advantage of this hierarchy: while working at the fine level we compute the search direction based on a coarse model. Importing concepts from information geometry to the n-orthotope, we propose a smooth...

Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al. (Math Program 129(1):33–68, 2011) to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction o...

Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impress...

Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impress...

Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al.\ \cite{kim2011exploiting} to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new de...

One of the challenges encountered in optimization of mechanical structures, in particular in what is known as topology optimization, is the size of the problems, which can easily involve millions of variables. A basic example is the minimum compliance formulation of the variable thickness sheet (VTS) problem, which is equivalent to a convex problem...

We formulate an inverse optimal design problem as a Mathematical Programming problem with Equilibrium Constraints (MPEC). The equilibrium constraints are in the form of a second-order conic optimization problem. Using the so-called Implicit Programming technique, we reformulate the bilevel optimization problem as a single-level nonsmooth nonconvex...

A small improvement in the structure of the material could save the manufactory a lot of money. The free material design can be formulated as an optimization problem. However, due to its large scale, second-order methods cannot solve the free material design problem in reasonable size. We formulate the free material optimization (FMO) problem into...

An interior point method for the structural topology optimization is proposed. The linear systems arising in the method are solved by the conjugate gradient method preconditioned by geometric multigrid. The resulting method is then compared with the so-called optimality condition method, an established technique in topology optimization. This metho...

The aim of this paper is to design an efficient multigrid method for constrained convex optimization problems arising from discretization of some underlying infinite dimensional problems. Due to problem dependency of this approach, we only consider bound constraints with (possibly) a single equality constraint. As our aim is to target large-scale p...

The discretization of constrained nonlinear optimization problems arising in the ?eld of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological ill-conditioning, strong nonlinearity and size. In this work we propose a methodology which brings together existing fast algorithms, namely, interior...

In this paper, we aim to solve the system of equations governing linear elasticity in parallel using domain decomposition. Through a non-overlapping decomposition of the domain, our approach aims to target the resulting interface problem, allowing for the parallel computation of solutions in an efficient manner. As a major application of our work,...

The connection between the sparsest solution to an underdetermined system of linear equations and the weighted ℓ1-minimization problem is established in this paper. We show that seeking the sparsest solution to a linear system can be transformed to searching for the densest slack variable of the dual problem of weighted ℓ1-minimization with all pos...

We propose a nonlinear domain decomposition algorithm for a class of nonlinear scalar PDEs which involves a sequence of nonlinear subdomain problems together with an interface problem. It is demonstrated that, when coupled with an appropriate preconditioner, the proposed method is able to compete with the commonly used Newton–Krylov method, deliver...

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 sam...

We propose a new algorithm for the solution of the robust multiple-load topology optimization problem. The algorithm can be applied to any type of problem, e.g., truss topology, variable thickness sheet or free material optimization. We assume that the given loads are uncertain and can be subject to small random perturbations. Furthermore, we defin...

This article is a continuation of the paper Kočvara and Stingl (Struct Multidisc Optim 33(4–5):323–335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems with local stress constraints. In particular, we consider the topology optimization (variable thickness sheet or “free sizing”) and...

The goal of this paper is to present an overview of the software collection for the solution of linear and nonlinear semidefinite optimization problems PENNON. In the first part we present theoretical and practical details of the underlying algorithm and several implementation issues. In the second part we introduce the particular codes PENSDP, PEN...

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 varia...

We propose a new look at the problem of truss topology optimization with integer or binary variables. We show that the problem can be equivalently formulated as an integer linear semidefinite optimization problem. This makes its numerical solution much easier, compared to existing approaches. We demonstrate that one can use an off-the-shelf solver...

A new method and algorithm for the efficient solution of a class of nonlinear semidefinite programming problems is introduced. The new method extends a concept proposed recently for the solution of convex semidefinite programs based on the sequential convex programming (SCP) idea. In the core of the method, a generally non-convex semidefinite progr...

The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a non- linear sem...

We study the discretized problem of the shape optimization of three-dimensional (3D) elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems obeying the Coulomb friction law. Mathematical modeling of the Coulomb friction problem leads to an implicit variational inequality. It is shown that for sma...

We present a compact overview of the recent development in free material optimization (FMO), a branch of structural optimization. The goal of FMO is to design the ultimately best material (its me-chanical properties and distribution in space) for a given purpose. We show that the current FMO mod-els naturally lead to linear and nonlinear semidefini...

Free material design deals with the question of finding the lightest structure subject to one or more given loads when both the distribution of material and the material itself can be freely varied. We additionally consider constraints on displacements of the optimal structure.

A new method for the efficient solution of free material optimization problems is in-troduced. The method extends the sequential convex programming (SCP) concept to a class of optimization problems with matrix variables. The basic idea of the new method is to approximate the original optimization problem by a sequence of sub-problems, in which nonl...

The paper considers a classic formulation of the topology optimization problem of discrete or discretized structures. The objective function to be maximized is the smallest natural frequency of the structure. We develop non-heuristic mathematical models paying special attention to the situation when some design variables take zero values. These mod...

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 t...

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 metho...

Free material design deals with the question of finding the lightest structure subject to one or more given loads when both the distribution of material and the material itself can be freely varied. We additionally consider constraints on local stresses in the optimal structure. We discuss the choice of formulation of the problem and the stress con...

The paper considers different problem formulations of topology optimization of discrete or discretized structures with eigenvalues as constraints or as objective functions. We study multiple-load case formulations of minimum volume or weight, minimum compliance problems, and the problem of maximizing the minimal eigenvalue of the structure, includi...

The goal of this paper is to formulate and solve free material optimization problems with constraints on the minimal eigenfrequency of a structure. A natural formulation of this problem as linear semidefinite program turns out to be numerically intractable. As alternative, we propose a new approach, which is based on a nonlinear semidefinite low-ra...

Delamination of elastic bodies glued together by an adhesive absorbing a specific amount of energy during the delamination process is formulated as an activated, rate-independent 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...

We present a new formulation of the truss topology problem that results in unique design and unique displacements of the optimal truss. This is reached by adding an upper level to the original optimization problem and formulating the new problem as an MPCC (Mathematical Program with Complementarity Constraints). We derive optimality conditions for...

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 pa...

Optimal control of magnetization in a ferromagnet is formulated as a mathematical program with evolutionary equilibrium constraints. To this purpose, we construct an evolutionary infinite-dimensional model which is discretized both in the space as well as in time variables. The evolutionary nature of this equilibrium is due to the hysteresis behavi...

We propose a new formulation of the worst-case multiple-load problem of free material optimization. It leads to an optimization problem with bilinear matrix inequality constraints. The resulting problem can be solved by a recently developed code PENBMI. The new formulation is shown to be more computationally efficient than the recently used one.

We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPleib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the publicly available general purpose software PENBMI. Numerical...

This paper is motivated by problem of optimal shape design of lami-nated elastic bodies. We use a recently introduced model of delamina-tion, based on minimization of potential energy which includes the free (Gibbs-type) energy and (pseudo)potential of dissipative forces, to in-troduce and analyze a special mathematical program with equilibrium con...

We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPleib . The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the publicly available general purpose software PENBMI. Numerica...

The goal of this article is to formulate and solve structural optimization problems with constraints on the global stability of the structure. The stability constraint is based on the linear buckling phenomenon. We formulate the problem as a nonconvex semidefinite programming problem and introduce an algorithm based on the augmented Lagrangian meth...

We consider a class of optimization problems with a generalized equation among the constraints. This class covers several problem types like MPEC (Mathematical Programs with Equilibrium Constraints) and MPCC (Mathematical Programs with Complementarity Constraints). We briefly review techniques used for numerical solution of these problems: penalty...

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-S...

This article describes a generalization of the PBM method by Ben-Tal and Zibulevsky to convex semidefinite programming problems. The algorithm used is a generalized version of the Augmented Lagrangian method. We present details of this algorithm as implemented in a new code PENNON. The code can also solve second-order conic programming (SOCP) probl...

The paper deals with a discretized problem of the shape optimization of elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems following the Coulomb friction law. Mathematical modelling of the Coulomb friction problem leads to a quasi-variational inequality. It is shown that for small coefficient...

A preconditioned conjugate gradient (PCG) method that is most suitable for reanalysis of structures is developed. The method presented provides accurate results efficiently. It is easy to implement and can be used in a wide range of applications, including non-linear analysis and eigenvalue problems. It is shown that the PCG method presented and th...

The goal of this paper is to find a computationally tractable formulation of the optimum truss design problem involving a constraint on the global stability of the structure. The stability constraint is based on the linear buckling phenomenon. We formulate the problem as a nonconvex semidefinite programming problem and briefly discuss an interior p...

Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material as the material itself can be freely varied. We present the single and multiple-load situation (understood in the worst-case sense). We further introduce a software tool MOPE...

An important objective of the study of mathematics is to analyze and visualize phenomena of nature and real world problems for its proper understanding. Gradually, it is also becoming the language of modem financial instruments. To project some of these developments, the conference was planned under the joint auspices of the Indian Society of Indus...

This paper presents a collection of tools for conceptual structure design. The underlying model is the 'free material optimization' problem. This problem gives the best physically attainable material and is considered as a generalization of the sizing/shape optimization problem. The method is supported by powerful optimization and numerical techniq...

Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material and the material itself can freely vary. The case of one single load has been discussed in several recent papers, and an efficient numerical approach was presented in [M. Kov...

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 numeri...

This paper deals with a central question of structural optimization: design of the stiffest structure occupying some fixed domain which is capable of carrying a given set of external loads. The design variables are the material properties at each point of the structure. In addition, we require that the structure can withstand small incidental force...

In this paper we discuss the application of a very efficient algorithm proposed recently by Kočvara and Zowe to American option pricing. Modelling and numerical simulation of options depending on the history of underlying asset price, inflation and devaluation by evolution equations and inequalities with hysteresis are proposed.

Control of systems can be performed directly in open loops in such a way that a certain criterium attains its optimal value. Typical examples of such criteria are cost, consumed energy, or total time to be minimized.

Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material and the material itself can freely vary. The case of one single load has been discussed in several recent papers, and an efficient numerical approach was presented in [M. Koc...

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, e...

. Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material and the material itself can be freely varied. We consider here the general multiple-load situation. After a series of transformation steps we reach a problem formulation fo...

We formulate two problems of optimal design for mechanical structures in unilateral contact: the truss topology problem and the material design problem for elastic bodies. In both cases we consider general multi-load formulations, where for each load-case we may have different set of contact constraints (rigid obstacles). We show that both problems...

In this chapter we use directional derivatives (Section 6.1) and generalized Jacobians (Section 6.2) to describe the local behaviour of the solution maps S for the perturbed generalized equations from Chapter 5. We restrict ourselves to the case of polyhedral feasible sets and assume further that the strong regularity condition holds. Similarly as...

As we will see later, variational inequalities (and complementarity problems) provide a convenient and elegant tool for characterizing manifold equilibria. The aim of this chapter is to spell out how these models can be brought into the equally useful form of a generalized equation $$ 0 \in C\left( z \right) + {N_Q}\left( z \right),$$ (4.1) where C...

Over the past twenty years a comprehensive theory has been developed on the stability of solutions to perturbed generalized equations, in particular on the Lipschitz behaviour of these solutions. In this chapter we present the part of this theory relevant to our later applications. We assume that the generalized equations from the previous chapter...

This chapter collects some nontrivial results which will be needed in the rest of the book. We try to keep the presentation as self-contained as possible.

Although the membrane problems discussed in the previous chapter have practical applications, they are usually considered as only model problems. Truly interesting practical examples arise from modelling of two- and three-dimensional elastic bodies. Their behaviour is again described by elliptic variational equations or inequalities.

Central to many aspects of economic analysis (and even political theory) is the concept of equilibrium. Similarly as in mechanics, formal characterizations of this concept typically come as complementarity problems or variational inequalities.

In our applications we have to deal with two types of nonsmooth problems: the minimization of a nonsmooth functional f and the solution of nonsmooth equations. This chapter presents in short two basic algorithms which can cope with the difficulty caused by the nondifferentiability. These two codes are working horses in the numerical part of this bo...

In this chapter we apply the preceding theory to establish first-order necessary optimality conditions and to construct an efficient and robust numerical method for the solution of the considered MPECs. This numerical method will extensively be used in the second (“applied”) part of the book.

In the previous two chapters we have introduced problems with obstacles and the problem of linear elasticity. The goal of this chapter is to combine those two and to introduce probably the most important ”nonsmooth“ problem of mechanics: the problem of an elastic body in contact with a rigid obstacle. A simple way how to define this problem is to r...