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Koc:92b_abstract: M. Kocvara. An iterative method for adaptive finite element computation. Computer Methods in Applied Mechanics and Engineering, 101:433--442, 1992.; Abstract. An iterative method designed for solving systems of equations arising from adaptive finite element computation is presented. The method is based on the block Gauss-Seidel algorithm. The convergence theory presented in the paper gives very sharp convergence bounds, which can easily be evaluated. The method applies for h-,p-extensions and various types of elements.
Koc:92a_abstract: M. Kocvara. An algebraic study of a local multigrid method for variational problems. Applied Mathematics and Computation, 51:17--41, 1992.; Abstract. A two-level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed. The smoothing steps consist of Gauss-Seidel relaxations applied only to the local mesh. A direct solver is required on the global coarse mesh. We analyze this scheme by modifying the standard algebraic convergence theory. The estimates of the convergence factor do not require any regularity assumption. The estimates can be evaluated element-by-element. The evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples. The theory also applies to a method with a direct solver on the local fine mesh. In this case we have obtained very sharp (and in fact optimal) estimates of the convergence factor.
Koc:93_abstract: M. Kocvara. An adaptive multigrid technique for three-dimensional elasticity. International Journal for Numerical Methods in Engineering, 36:1703--1716, 1993.; Abstract. A program for finite element analysis of 3D linear elasticity problems is described. The program uses quadratic hexahedral elements. The solution process starts on an initial coarse mesh; here error estimators are determined by the standard Babuska-Rheinboldt's method and local refinement is performed by partitioning of indicated elements, each hexahedron into eight new elements. Then the discrete problem is solved on the second mesh and the refinement process proceeds in the following way --- on the i-th mesh only the elements caused by refinement on the (i-1)-th mesh can be refined. The control of refinement is the task of the user because the dimension of the discrete problem grows very rapidly in 3D. The discrete problem is being solved by the frontal solution method on the initial mesh and by a newly developed and very efficient local multigrid method on the refined meshes. The program can be successfully used for solving problems with structural singularities, such as re-entrant corners and moving boundary conditions. A numerical example shows that such problems are solved with the same efficiency as regular problems are.
KZ:94_abstract: M. Kocvara, J. Zowe. An iterative two-step method for linear complementarity problems. Numerische Mathematik, Vol 68 (1994), 95--106.; Abstract. We propose an algorithm for the numerical solution of large-scale symmetric positive-definite linear complementarity problems. Each step of the algorithm combines an application of the successive overrelaxation method with projection (to determine an approximation of the optimal active set) with the preconditioned conjugate gradient method (to solve the reduced residual systems of linear equations). Convergence of the iterates to the solution is proved. In the experimental part we compare the efficiency of the algorithm with several other methods. As test example we consider the obstacle problem with different obstacles. For problems of dimension up to 24\,000 variables, the algorithm finds the solution in less then 7 iterations, where each iteration requires about 10 matrix-vector multiplications.
KO:94a_abstract: M. Kocvara and J. V. Outrata. On optimization of systems governed by implicit complementarity problems. Numerical Functional Analysis and Optimization, 15:869--887, 1994.; Abstract. We consider a class of parameter-dependent implicit complementarity problems possessing for each value of the parameter (control) from a given set a unique solution. Then an optimization problem can be formulated in which such an implicit complementarity problem arises as a constraint. We analyze these optimization problems with the tools of nonsmooth analysis and propose an approach to their numerical solution, using a bundle method from nondifferentiable optimization and a nonsmooth variant of the Newton method. As a test example, the so-called packaging problem, known from the optimum shape design is taken, in which, however, the standard rigid obstacle is replaced by an elastic one.
KO:95a_abstract: M. Kocvara and J. V. Outrata. On the solution of optimum design problems with variational inequalities. In: Recent Advances in Nonsmooth Optimization (D.-Z. Du, L. Qi and R.L. Womersley, eds.), pages 171--191. World Scientific Publishers, 1995.; Abstract. The paper deals with the numerical solution of a class of optimum design problems in which the controlled systems are described by elliptic variational inequalities. The approach is based on the characterization of (discretized) system operators by means of generalized Jacobians and the subsequent usage of nondifferentiable optimization methods. As an application, two important shape design problems are solved.
KO:95b_abstract: M. Kocvara and J. V. Outrata. On a class of quasi-variational inequalities. Optimization Methods and Software, 5:275--295, 1995.; Abstract. 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.
KZ:98_abstract: M. Kocvara and J. Zowe. Free Material Optimization. Documenta Mathematica J. DMV, Extra Volume ICM 1998, III, 707-716.; Abstract. 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 for which we can prove existence of a solution; a suitable discretization leads to a semidefinite programming problem for which modern polynomial time algorithms of interior-point type are available. Two numerical examples demonstrate the efficiency of our approach.
KZN:98_abstract: M. Kocvara, J. Zowe and A. Nemirovski. Cascading-An Approach to Robust Material Optimization. NATO ASI Mechanics of Composite Materials and Structures, Troia, Portugal, July 12-24, 1998.; Abstract. 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 forces which are not known a-priori. We introduce the notion of robust design and propose a solution technique called cascading. This technique enables us to find a good approximation of the robust design with a relatively small computational effort. Examples demonstrate efficiency of this technique and importance of the robust structural design itself.
BJKNZ:00_abstract: A. Ben-Tal, F. Jarre, M. Kocvara, A. Nemirovski and J. Zowe. Optimal design of trusses under a nonconvex global buckling constraint.; Abstract. 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.
SMK:00_abstract: A. H. Siddiqi, P. Manchanda, and M. Kocvara. An Iterative Two-Step Algorithm for American Option Pricing.; Abstract. In this paper we discuss the application of a very efficient algorithm proposed recently by Ko\v{c}vara 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.
Koc:91_abstract: M. Kocvara. LAME --- local adaptive multigrid for 3D elasticity. In Proc. of the VI. Internat. Conf. on Mathematical Methods in Engineering, Plzen, 1991.; Abstract. LAME is a program for solving 3D linear elasticity problems by quadratic finite elements. The solution process starts on an initial coarse mesh; here the error estimators are determined and local mesh refinement is performed. The discrete problem is solved on the second mesh and the refinement prosess proceed --- control of refinement is left to the user. Corresponding systems of algebraic equations are solved by a newly-developed multigrid algorithm. The algorithm is described in the paper as well as results of an illustrative numerical experiment.
KO:93_abstract: M. Kocvara and J. V. Outrata. On implicit complementarity problems with application in mechanics. In J.-P. Zolésio, editor, Proc. of the IFIP Conf. on Numerical Analysis and Optimization, Rabat, Dec. 15--17 1993. To appear.; Abstract. We consider a class of parameter-dependent implicit complementarity problems possessing for each value of the parameter (control) from a given set a unique solution. Then an optimization problem can be formulated in which such an implicit complementarity problem arises as a constraint. We analyze these optimization problems with the tools of nonsmooth analysis and propose an approach to their numerical solution, using a bundle method from nondifferentiable optimization and a nonsmooth variant of the Newton method. As a test example, the so-called packaging problem, known from the optimum shape design is taken, in which, however, the standard rigid obstacle is replaced by an elastic one.
KZ:95a_abstract: M. Kocvara and J. Zowe. How to optimize mechanical structures simultaneously with respect to topology and geometry. Proceedings of the First World Congress on Structural and Multidisciplinary Optimization, Goslar, Germany, May 28--June 2, 1995. To appear.; Abstract. We present the numerical progress in an area which we have already addressed in a paper by Ben-Tal, Kocvara and Zowe (1993). Now, due to substantial software improvements, we can optimize at the same time the topology and the geometry of realistic discrete three-dimensional structures (trusses) on a simple workstation. For almost all test examples our concept resulted in more sparse structures and in better compliance values (in less CPU-time) than a ground-structure/topology approach, which imitates the geometry aspect by starting from a very dense mesh of potential nodes and bars. An especially attractive feature of our approach lies in the fact that we also allow moves of nodes under load and even of nodes which fix the structure to a rigid support. Note that this is impossible in the ground-structure/topology approach. Thus the question of shape optimization is also addressed in our concept. Moreover, our approach can deal with the optimization of the shape of continuum structures in the framework of material optimization. Realistic examples will demonstrate the efficiency of our concept.
KZ:96_abstract: M. Kocvara and J. Zowe. How mathematics can help in design of mechanical structures. Proceedings of the 16th Biennial Conference on Numerical Analysis, Dundee, June 27--30, 1995. To appear.; Abstract. Structural optimization combines elements from mathematics and from engineering in a challenging way. This paper is written with the goal of convincing the reader that this fascinating area deserves closer cooperation from researchers of both disciplines and that such teamwork can lead to substantial progress in this field. The first section briefly presents the mathematical-mechanical model of the structures we are interested in. To avoid merely technical difficulties, we keep this model as simple as possible. However, even in this slimmed-down version the mathematical formulation is not accessible to standard optimization software. This is mainly due to the fact that the design variables fall into two groups, which differ fundamentally in character, and which are hard to treat in parallel. Hence engineers resort here to more or less dubious heuristic approaches. In what follows, we will show how to overcome these difficulties and how to solve the problem in an efficient and rigorous mathematical way. The first approach eliminates one block of the design variables at the price of a drastic increase in the dimension of the remaining block of unknowns. A thorough analysis of the resulting problem leads to a reformulation as a simple constrained problem in the state variables only, with the design variables as Lagrange multipliers. This problem can be attacked successfully by modern interior-point methods. The second approach works with a natural grouping of the design variables into two blocks, each of which, taken on its own, is full of structure. We try to maintain this structure by separating the variables and dealing with them on two levels of hierarchy. The price for the gain in structure is the introduction of nonsmoothness on the upper level of the problem. Hence nonsmooth methods are required here. Such software exists and can deal well with the nondifferentiability in question. A series of examples will demonstrate the efficiency of the two approaches.
KO:97a_abstract: M. Kocvara and J. V. Outrata. A Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. In M. Ferris and J.-S. Pang, editors, Proceedings of the International Conference on Complementarity Problems, Baltimore, November 1--4, 1995. To appear.; Abstract. We consider a class of optimization problems in which either nonlinear complementarity or implicit complementarity problems arise as side constraints. Under the assumption of strong regularity of Robinson at a local minimum, we derive necessary optimality conditions and propose a numerical approach to the solution of these problems. This approach is applied to a design problem from continuum mechanics.
JKZ:98_abstract: F. Jarre, M. Kocvara and J. Zowe. Interior point methods for mechanical design problems.; Abstract. This article presents a primal-dual predictor-corrector interior-point method for solving quadratically constrained convex optimization problems that arise from truss design. We investigate certain special features of the problem, discuss fundamental differences of interior-point methods for linear programs and nonlinearly constrained problems, and extend Mehrotra's predictor-corrector strategy to nonlinear programs. Numerical experiments on large scale problems illustrate the surprising efficiency of the method.
ZKB:97_abstract: J. Zowe, M. Kocvara and M. Bendsoe. Free Material Optimization via Mathematical Programming.; Abstract. 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.
Koc:97_abstract: M. Kocvara. Topology Optimization with Displacement Constraints: A Bilevel Programming Approach.; Abstract. We consider the minimum-compliance formulation of the truss topology problem with additional linear constraints on the displacements: the so-called displacement constraints. We propose a new bilevel programming approach to this problem. Our primal goal (upper-level) is to satisfy the displacement constraint as well as possible --- we minimize the gap between the actual and prescribed displacement. Our second goal (lower-level) is to minimize the compliance --- we still want to find the stiffest structure satisfying the displacement constraints. On the lower level we solve a standard truss topology problem and hence we can solve it in the formulation suitable for the fast interior-point algorithms. The overall bilevel problem is solved by means of the so-called implicit programming approach. This approach leads to a nonsmooth optimization problem which is finally solved by a nonsmooth solver.
KZZ:98_abstract: M. Kocvara, M. Zibulevsky and J. Zowe. Mechanical design problems with unilateral contact.; Abstract. 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 (after discretization of the latter one) can be rewritten as mathematical programs, which only differ in the character of the input data but otherwise have identical structure and thus allow the same algorithmic approach. We propose an iterative optimization algorithm based on penalty-barrier methods. A series of numerical examples demonstrates the usability and efficiency of our approach.
BKNZ:99_abstract: A. Ben-Tal, M. Kocvara, A. Nemirovski and J. Zowe. Free Material Design via Semidefinite Programming. The Multi-Load Case with Contact Conditions.; Abstract. 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. The case of one single load has been discussed in several recent papers and an efficient numerical approach was presented in \cite{kocvara-zibulevsky-zowe}. We attack here the multi-load situation (understood in the worst-case sense) which is of much more interest for applications but also significantly more challenging, both from the theoretical and the numerical point of view. After a series of transformation steps we reach a problem formulation for which we can prove existence of a solution; a suitable discretization leads to a semidefinite programming problem for which modern polynomial time algorithms of interior-point type are available. A number of numerical examples demonstrates the efficiency of our approach.
Koc:00_abstract: M. Kocvara. Modern Optimization Algorithms in Topology Design.; Abstract. The lecture consists of two parts. First we introduce the latest most powerful methods of numerical optimization, interior point and penalty methods. By using simple examples and (especially) pictures we outline the basic ideas behind the efficiency of the methods. We also introduce the notion of convex optimization problem and show its importance for the efficiency of optimization algorithms. The second part of the lecture is focused on practical applications of the above methods. We present a software system MOPED (Material Optimization with PEnalty methoD) and demonstrate how a highly efficient tool, able to solve very large problems, can be devised when a proper (convex) mathematical formulation of the problem is connected with the powerful penalty/barrier method. Further, on a particular example of optimum design with stability constraints, we show that even a nonconvexity in the problem formulation is not a serious obstacle to an interior point algorithm.
KSW:00_abstract: M. Kocvara, M. Stingl, and R. Werner. MOPED User's Guide. Version 1.02.; Abstract. MOPED stands for Material Optimization In Engineering Design. This software package computes optimal material distribution/properties in an arbitrary two-dimensional domain under single-load condition. The triangulation of the domain is done by the code \domesh. The optimization problem is solved by a penalty/barrier method due to M. Zibulevsky. We give instructions how to install and use the code.
HKW:01_abstract: H.R.E.M. Hornlein, M. Kocvara, and R. Werner. MOPED---An Integrated Designer Tool for Material Optimization. (Material Optimization: Bridging the Gap between Conceptual and Preliminary Design); Abstract. This paper presents a collection of tools for conceptual structure design, called MOPED. The underlying model is the free material optimization problem. This problem gives the best physically attainable material and can be considered the ``ultimate'' generalization of the structural/shape optimization problem. The method is supported by powerful optimization and numerical techniques, which allow us to work with bodies of complex initial design and with very fine finite-element meshes, giving thus quite accurate solutions even for bodies with complex geometry. The computed results are realized by composite materials. We consider a particular class of fibre-reinforced composite materials manufactured by the so-called tape-laying technology. In the post-processing phase, we plot curves which indicate how to lay these tapes; they also show the proposed thickness of the tapes. Several examples demonstrate that MOPED is an ideal tool for conceptual design of engineering structures.
OKZ:98_abstract: J. Outrata, M. Kocvara and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results; Abstract. This book presents an in-depth study and a solution technique for an important class of optimization problems. This class is characterized by special constraints: parameter-dependent convex programs, variational inequalities or complementarity problems. All these so-called equilibrium constraints are mostly treated in a convenient form of generalized equations. The book begins with a chapter on auxiliary results followed by a description of main numerical tools:a bundle method of nonsmooth optimization and a nonsmooth variant of Newton's method. Following this, stability and sensitivity theory for generalized equations is presented, based on the concept of strongi regularity. This enables to apply the generalized differential calculus for Lipschitz maps to derive optimality conditions and to arrive at a solution method. A large part of the book focuses on applications coming from continuum mechanics and mathematical economy. A series of nonacademic problems is introduced and analyzed in detail. Each problem is accompanied with examples that show the efficiency of the solution method. This book is addressed to applied mathematicians and engineers working in continuum mechanics, operations research and economic modelling. Students interested in optimization will find the book useful as well.
KKZ:02_abstract: U. Kirsch, M. Kocvara, and J. Zowe. Accurate Reanalysis of Structures by a Preconditioned Conjugate Gradient Method.; Abstract. 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 nonlinear analysis and eigenvalue problems. It is shown that the PCG method presented and the Combined Approximations (CA) method developed recently provide theoretically identical results. Consequently, available results from one method can be applied to the other method. Effective solution procedures developed for the CA method can be used for the PCG method, and various criteria and error bounds developed for Conjugate Gradient (CG) methods can be used for the CA method. Numerical examples show that the condition number of the selected preconditioned matrix is much small er than the condition number of the original matrix. This property explains the fast convergence and accurate results achieved by the method.
BHKO:01_abstract: P. Beremlijski, J. Haslinger, M. Kocvara, and J. Outrata. Shape Optimization in Contact Problems with Coulomb Friction.; Abstract. The paper deals with a discretized problem of shape optimization of elastic bodies in unilateral contact. The aim is to extend the existing results to the case of contact problems following the Coulomb friction law. Mathematical model of the Coulomb friction problem leads to a quasi-variational inequality. It is shown that for small coefficients of friction, the discretized problem with Coulomb friction has a unique solution and that this solution is Lipschitzian as a function of a control variable, describing the shape of the elastic body. The shape optimization problem belongs to a class of so-called mathematical programs with equilibrium constraints (MPECs). The uniqueness of the equilibria for fixed controls enables us to apply the so-called implicit programming approach. Its main idea consists in minimization of a nonsmooth composite function generated by the objective and the (single-valued) control-state mapping. In this paper, the control-state mapping is much more complicated than in most MPECs solved so far in the literature, and the generalization of the relevant results is by no means straightforward. Numerical examples illustrate the efficiency and reliability of the suggested approach.
KS:01_abstract: M. Kocvara and M. Stingl. PENNON - A Generalized Augmented Lagrangian Method for Semidefinite Programming.; Abstract. 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) problems, as well as problems with a mixture of SDP, SOCP and NLP constraints. Results of extensive numerical tests and comparison with other SDP codes are presented.
KS:02_abstract: M. Kocvara and M. Stingl. PENNON - A Code for Convex Nonlinear and Semidefinite Programming.; Abstract. 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.
Koc:02_abstract: M. Kocvara. On the modelling and solving of the truss design problem with global stability constraints.; Abstract. 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 point technique for the numerical solution of this problem. We further discuss relation to other models. The paper is concluded by a series of numerical examples.
KO:03_abstract: M. Kocvara and J. Outrata. Effective reformulations of the truss topology design problem.; Abstract. 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 this problem and present several techniques for its numerical solution. Finally, we compare two of these techniques on a series of numerical examples.
KO:03_abstract: M. Kocvara and J. Outrata. Optimization problems with equilibrium constraints and their numerical solution.; Abstract. 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 methods, nonlinear programming (NLP) techniques and Implicit Programming approach (ImP). We further present a new theoretical framework for the ImP technique that is particularly useful in case of difficult equilibria. Finally, three numerical examples are presented: an MPEC that can be solved by ImP but can hardly be formulated as a nonlinear program, an MPCC that cannot be solved by ImP and finally an MPEC solvable by both, ImP and NLP techniques. In the last example we compare the efficiency of the two approaches.
KO:03b_abstract: M. Kocvara and J. Outrata. On the modeling and control of delamination processes.; Abstract. This paper is motivated by problem of optimal shape design of laminated elastic bodies.
We use a recently introduced model of delamination, based on minimization of potential energy which includes the free
(Gibbs-type) energy and (pseudo)potential of dissipative forces, to introduce and analyze a special mathematical program with equilibrium constraints.
The equilibrium is governed by a finite sequence of coupled mathematical programs that have to be solved one after another in the direction of increasing time.
We derive optimality conditions for the control problem and illustrate them on an academic example.
HKS:03_abstract: D. Henrion, M. Kocvara, M. Stingl. Solving simultaneous stabilization BMI problems with PENNON.; Abstract. A class of iterative methods for convex nonlinear programming problems was introduced by Ben-Tal and Zibulevsky and named PBM. The framework of the algorithm is given by the augmented Lagrangian method; the difference to the classic
algorithm is in the definition of a special penalty/barrier function satisfying certain properties. A generalization of the PBM method for convex semidefinite programming problems was recently proposed by Kocvara and Stingl. The algorithm was implemented in the code PENNON, that proved to be very efficient for linear SDP problems. Recently, the algorithm has been generalized to nonlinear semidefinite programming problems. In this talk, the resulting algorithm is applied to a special class of nonlinear semidefinite programming problems, where a linear objective is minimized with respect to bilinear matrix inequalities (BMI). We will present numerical results of the method for a class of optimization problems coming from control theory, the simultaneous stabilization problem. Simultaneous stabilization consists in finding one unique controller that stabilizes a set of given linear plants. This problem arises when seeking a robust control law for systems potentially subject to actuator or sensor failures. Following a pure algebraic/polynomial approach, the simultaneous stabilization problem can be formulated as a BMI problem in the parameters of the controller, whose order can be fixed from the outset. This is in stark contrast with other approaches to simultaneous stabilization, for which it is very often difficult to bound the order of the controller. Another advantage of the polynomial formulation over the (more classical) state-space formulation for this robust control problem is that there is no need to seek a Lyapunov matrix certifying stability. The number of decision variables in the design BMI is then drastically reduced.
KS:04_abstract: M. Kocvara and M. Stingl. Solving Nonconvex SDP Problems of Structural Optimization with Stability Control.; Abstract. The goal of this paper 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 method combined with the Trust-Region technique. The algorithm is implemented in a code PENNON. The paper is concluded by a series of numerical examples.
KLSH:04_abstract: M. Kocvara, F. Leibfritz, M. Stingl, and D. Henrion. A nonlinear SDP algorithm for static output feedback problems in COMPlib.; Abstract. 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 COMPlib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the publicly available general purpose software PENBMI. Numerical results demonstrate the behavior of the code.
KS:05_abstract: M. Kocvara and M. Stingl. On the solution of large-scale SDP problems by the modified barrier method using iterative solvers.; Abstract.When solving large-scale semidefinite programming problems by second-order methods, the storage and factorization of the Newton matrix are the limiting factors. 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 large number of variables and modest size of the constrained matrix. We further propose to approximate the Newton matrix in the matrix-vector product by a finite-difference formula. This leads to huge savings in memory requirements and, for certain problems, to further speed-up of the algorithm.
KKO:05_abstract: M. Kocvara, M. Kruzik and J.V. Outrata. On the control of an evolutionary equilibrium in micromagnetics.; Abstract. We formulate an optimal control problem of magnetization in a ferromagnet as a mathematical program with evolutionary equilibrium constraints. The evolutionary nature of the equilibrium is due to the hysteresis behavior of the respective magnetization process. To solve the problem numerically, we adapted the implicit programming technique. The adjoint equations, needed to compute the subgradients of the composite objective, are derived using the Mordukhovich's generalized differential calculus. We show the existence of a solution to such program and discuss results of computational experiments.
AK:06_abstract: W. Achtziger and M. Kocvara. Structural Topology Optimization with Eigenvalues.; Abstract. 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 weight, minimum compliance problems and of the problem of maximizing the minimal eigenvalue of the structure including the effect of non-structural mass. The paper discusses interrelations of the problems and, in particular, shows how solutions of one problem can be derived from solutions of the other ones. Moreover, we present equivalent reformulations as semidefinite programming problems with the property that, for the minimum weight and minimum compliance problem, each local optimizer of these problems is also a global one. This allows for the calculation of guaranteed global optimizers of the original problems by the use of modern solution techniques of semidefinite programming. For the problem of maximization of the minimum eigenvalue we show how to verify the global optimality and present an algorithm for finding a tight approximation of a globally optimal solution. Numerical examples are provided for truss structures. Examples of both academic and larger size illustrate the theoretical results achieved and demonstrate the practical use of this approach. We conclude with an extension on multiple non-structural mass conditions.
SKL:07a_abstract: M. Stingl, M. Kocvara, and G. Leugering: A Sequential Convex Semidefinite Programming Algorithm for Multiple-Load Free Material Optimization; Abstract: A new method for the efficient solution of free material optimization problems is introduced. 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 subproblems, in which nonlinear functions (defined in matrix variables) are approximated by block-separable convex functions. The subproblems are semidefinite programs with a favorable structure, which can be efficiently solved by existing SDP software. The new method is shown to be globally convergent. The article is concluded by a series of numerical experiments demonstrating the effectiveness of the generalized SCP approach.
SKL:07a_abstract: M. Stingl, M. Kocvara, and G. Leugering: Free Material Optimization with Control of the fundamental Eigenfrequency; Abstract: 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-rank approximation of the semidefinite dual. Throughout this article, an algorithm is introduced and convergence properties are investigated. The article is concluded by numerical experiments proving the effectiveness of the new approach.
KSZ:08_abstract: M. Kocvara, M. Stingl, and J. Zowe: Free Material Optimization: Recent Progress; Abstract: 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 mechanical properties and distribution in space) for a given purpose. We show that the current FMO models naturally lead to linear and non-linear semidefinite programming problems (SDP); their numerical tractability is then guaranteed by recently introduced SDP algorithms.

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Michal Kocvara
March 9, 2008