Discretization in semi-infinite programming: the rate of convergence

The discretization approach for solving semi-infinite optimization problems is considered. We are interested in the convergence rate of the error between the solution of the semi-infinite problem and the solution of the discretized program depending on the discretization mesh-size. It will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a specific way. This is done for ordinary and for generalized semi-infinite problems. Received: November 21, 2000 / Accepted: May 2001?Published online September 17, 2001     

Properties of an interior embedding for solving nonlinear optimization problems

The paper presents an interior embedding of nonlinear optimization problems. This embedding satisfies a sufficient condition for the success of pathfollowing algorithms with jumps being applied to one-parametric optimization problems.?The one-parametric problem obtained by the embedding is supposed to be regular in the sense of Jongen, Jonker and Twilt. This asumption is analyzed, and its genericity is proved in the space of the original optimization problems. Received May 20, 1997 / Revised version received October 6, 1998?Published online May 12, 1999     

Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples. Received: October 1998 / Accepted: June 2000?Published online October 18, 2000     

On generalized trade-off directions in nonconvex multiobjective optimization

Trade-off information related to Pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives. Recently, the concept of trade-off directions has been introduced for convex problems. These trade-offs are characterized with the help of tangent cones. Generalized trade-off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones. Here we study how the convex concepts and results can be generalized into a nonconvex case. Giving up convexity naturally means that we need local instead of global analysis. Received: December 2000 / Accepted: October 2001?Published online February 14, 2002     

Forcing strong convergence of proximal point iterations in a Hilbert space

This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by Güler [11] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns. Received January 6, 1998 / Revised version received August 9, 1999?Published online November 30, 1999     

Shallow, deep and very deep cuts in the analytic center cutting plane method

In this paper we show that the cut does not need to go through the query point: it can be deep or shallow. The primal framework leads to a simple analysis of the potential variation, which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step. Received July 3, 1996 / Revised version received July 11, 1997 Published online August 18, 1998     

Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems

We are concerned with the problem of existence, uniqueness and qualitative properties of solutions to the radially symmetric variational problem where is the ball of centered at the origin and with radius , the map is a normal integrand, and is a convex function of the second variable. This kind of problems, with non-convex lagrangians with respect to , arise in various fields of applied sciences, such as optimal design and nonlinear elasticity. Received June 18, 1998; in final form August 26, 1999 / Published online September 14, 2000     

Well-posedness and porosity in optimal control without convexity assumptions

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In [27] we considered a class of optimal control problems which is identified with the corresponding complete metric space of integrands, say . We did not impose any convexity assumptions. The main result in [27] establishes that for a generic integrand the corresponding optimal control problem is well-posed. In this paper we study the set of all integrands for which the corresponding optimal control problem is well-posed. We show that the complement of this set is not only of the first category but also a -porous set. The main result of the paper is obtained as a realization of a variational principle which can be applied to various classes of optimization problems. Received April 15, 2000 / Accepted October 10, 2000 / Published online December 8, 2000     

Convex programming for disjunctive convex optimization

Given a finite number of closed convex sets whose algebraic representation is known, we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets. We derive an algebraic characterization of the feasible region in a higher-dimensional space and propose a solution procedure akin to the interior-point approach for convex programming. Received November 27, 1996 / Revised version received June 11, 1999?Published online November 9, 1999     

Parameter estimation in systems of nonlinear equations

Summary. A numerical method is presented to determine parameters in a system of non\-linear equations in the following sense: Proceeding from given experimental data, e.g., observation times and measurements, the minimum least-squares distance of the measured data from a fitting criterion depending on the solution of a system of nonlinear equations is to be computed. Specifically coupled mass equilibrium models are described in more detail that are used in pharmaceutical applications for receptor-ligand binding studies. They are used for instance for the radioimmunological determination of Fenoterol or related substances. Numerical results based on laboratory data are included to test the robustness of the algorithm implemented. Received May 24, 1993/Revised version received February 13, 1994     

Characterization of the smoothness and curvature of a marginal function for a trust-region problem

Received May 28, 1996 / Revised version received May 1, 1998 Published online October 9, 1998     

New properties of a nonlinear conjugate gradient method

Summary. This paper provides several new properties of the nonlinear conjugate gradient method in [5]. Firstly, the method is proved to have a certain self-adjusting property that is independent of the line search and the function convexity. Secondly, under mild assumptions on the objective function, the method is shown to be globally convergent with a variety of line searches. Thirdly, we find that instead of the negative gradient direction, the search direction defined by the nonlinear conjugate gradient method in [5] can be used to restart any optimization method while guaranteeing the global convergence of the method. Some numerical results are also presented. Received March 12, 1999 / Revised version received April 25, 2000 / Published online February 5, 2001     

On the equivalence of two variational problems

We consider the following problems where is a convex function, is an open bounded subset of is a closed convex subset of such that and and are suitable obstacles. We give conditions on the function {\it g} under which the two problems are equivalent. Received March 24, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Modeling languages and Condor: metacomputing for optimization

A generic framework is postulated for utilizing the computational resources provided by a metacomputer to concurrently solve a large number of optimization problems generated by a modeling language. An example of the framework using the Condor resource manager and the AMPL and GAMS modeling languages is provided. A mixed integer programming formulation of a feature selection problem from machine learning is used to test the mechanism developed. Due to this application’s computational requirements, the ability to perform optimizations in parallel is necessary in order to obtain results within a reasonable amount of time. Details about the simple and easy to use tool and implementation are presented so that other modelers with applications generating many independent mathematical programs can take advantage of it to significantly reduce solution times. Received: October 28, 1998 / Accepted: December 01, 1999?Published online June 8, 2000     

On regularity and stability in semi-infinite optimization

For stationary solutions and Langrange multipliers of a semi-infinite program withC ¹ data, we study some stability behaviour which is closely related to (metric) regularity of the constraint system. The multiplier set mapping considered here has its images in a finite-dimensional space. In this framework, regularity is a necessary and sufficient condition to have bounded sets of multipliers.     

Conditioning of semidefinite programs

theory. This approach also quantifies the size of permissible perturbations. We include a discussion of these results for block diagonal semidefinite programs, of which linear programming is a special case. Received November 26, 1995 / Revised version received November 1, 1998 Published online February 25, 1999     

Global s-type error bound for the extended linear complementarity problem and applications

For the extended linear complementarity problem over an affine subspace, we first study some characterizations of (strong) column/row monotonicity and (strong) R ₀-property. We then establish global s-type error bound for this problem with the column monotonicity or R ₀-property, especially for the one with the nondegeneracy and column monotonicity, and give several equivalent formulations of such error bound without the square root term for monotone affine variational inequality. Finally, we use this error bound to derive some properties of the iterative sequence produced by smoothing methods for solving such a problem under suitable assumptions. Received: May 2, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

Upper Lipschitz behavior of solutions to perturbed. C1,1 programs

We analyze the local upper Lipschitz behavior of critical points, stationary solutions and local minimizers to parametric C ^1,1 programs. In particular, we derive a characterization of this property for the stationary solution set map without assuming the Mangasarian–Fromovitz CQ. Moreover, conditions which also ensure the persistence of solvability are given, and the special case of linear constraints is handled. The present paper takes pattern from [21] by continuing the approach via contingent derivatives of the Kojima function associated with the given optimization problem. Received: June 10, 1999 / Accepted: November 15, 1999?Published online July 20, 2000     

Dual conditions characterizing optimality for convex multi-objective programs

Received June 13, 1995 / Revised version received February 6, 1998 Published online August 18, 1998