A modified standard embedding with jumps in nonlinear optimization

The paper deals with a combination of pathfollowing methods (embedding approach) and feasible descent direction methods (so-called jumps) for solving a non-linear optimization problem with equality and inequality constraints. Since the method that we propose here uses jumps from one connected component to another one, more than one connected component of the solution set of the corresponding one-parametric problem can be followed numerically. It is assumed that the problem under consideration belongs to a generic subset which was introduced by Jongen, Jonker and Twilt. There already exist methods of this type for which each starting point of a jump has to be an endpoint of a branch of local minimizers. In this paper the authors propose a new method by allowing a larger set of starting points for the jumps which can be constructed at bifurcation and turning points of the solution set. The topological properties of those cases where the method is not successful are analyzed and the role of constraint qualifications in this context is discussed. Furthermore,this new method is applied to a so-called modified standard embedding which is a particular construction without equality constraints. Finally, an algorithmic version of this new method as well as computational results are presented

Mathematical programs with complementarity constraints: convergence properties of a smoothing method

In this paper, optimization problems $P$ with complementarity constraints are considered. Characterizations for local minimizers $\bar{x}$ of $P$ of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which $P$ is replaced by a perturbed problem $P_{\tau}$ depending on a (small) parameter $\tau$. We are interested in the convergence behavior of the feasible set $\cal{F}_{\tau}$ and the convergence of the solutions $\bar{x}_{\tau}$ of $P_{\tau}$ for $\tau\to 0.$ In particular, it is shown that, under generic assumptions, the solutions $\bar{x}_{\tau}$ are unique and converge to a solution $\bar{x}$ of $P$ with a rate $\cal{O}(\sqrt{\tau})$. Moreover, the convergence for the Hausdorff distance $d(\cal{F}_{\tau}$, $\cal{F})$ between the feasible sets of $P_{\tau}$ and $P$ is of order $\cal{O}(\sqrt{\tau})$

On the stratification of a class of specially structured matrices

We consider specially structured matrices representing optimization problems with quadratic objective functions and (finitely many) affine linear equality constraints in an n-dimensional Euclidean space. The class of all such matrices will be subdivided into subsets ['strata'], reflecting the features of the underlying optimization problems. From a differential-topological point of view, this subdivision turns out to be very satisfactory: Our strata are smooth manifolds, constituting a so-called Whitney Regular Stratification, and their dimensions can be explicitly determined. We indicate how, due to Thom's Transversality Theory, this setting leads to some fundamental results on smooth one-parameter families of linear-quadratic optimization problems with ( finitely many) equality and inequality constraints

MULTI-OBJECTIVE EFFICIENT PARAMETRIC OPTIMIZATION

Parametric optimization is the process of solving an optimization problem as a function of currently unknown or changing variables known as parameters. Parametric optimization methods have been shown to benefit engineering design and optimal morphing applications through its specialized problem formulation and specialized algorithms. Despite its benefits to engineering design, existing parametric optimization algorithms (similar to many optimization algorithms) can be inefficient when design analyses are expensive. Since many engineering design problems involve some level of expensive analysis, a more efficient parametric optimization algorithm is needed. Therefore, the multi-objective efficient parametric optimization (MO-EPO) algorithm is developed to allow for the efficient optimization of problems with multiple parameters and objectives. This technique relies on the new parametric hypervolume indicator (pHVI) which measures the space dominated by a given set of data involving both objectives and parameters. The pHVI benefits parametric optimization by enabling the comparison of optimization results, enabling the visualization and detection of optimization convergence, and providing information for an optimization algorithm. MO-EPO uses response surface models of expensive functions to find and evaluate a designs expected to improve the solution and/or models. With new information, response surface models are updated and the process is repeated. "Improvement" is measured by the pHVI metric allowing for the consideration of any number of objectives and parameters. The novel MO-EPO algorithm is demonstrated on a number of analytical benchmarking problems and two distinct morphing applications with various numbers of objectives and parameters. In each case, MO-EPO is shown to find solutions that are as good as or better than those found from the existing P3GA (i.e., equal or greater pHVI value) when the number of design evaluations is limited. Both the pHVI metric and the MO-EPO algorithm are significant contributions to parametric optimization methodology and engineering design

Parametric programming: An illustrative mini encyclopedia

Parametric programming is one of the broadest areas of applied mathematics. Practical problems, that can be described by parametric programming, were recorded in the rock art about thirty millennia ago. As a scientific discipline, parametric programming began emerging only in the 1950\u27s. In this tutorial we introduce, briefly study, and illustrate some of the elementary notions of parametric programming. This is done using a limited theory (mainly for linear and convex models) and by means of examples, figures, and solved real-life case studies. Among the topics discussed are stable and unstable models, such as a projectile motion model (maximizing the range of a projectile), bilevel decision making models and von Stackelberg games of market economy, law of refraction and Snell\u27s law for the ray of light, duality, Zermelo\u27s navigation problems under the water, restructuring in a textile mill, ranking of efficient DMU (university libraries) in DEA, minimal resistance to a gas flow, and semi-abstract parametric programming models. Some numerical methods of input optimization are mentioned and several open problems are posed

Solving bilevel programs with the KKT-approach

Bilevel programs (BL) form a special class of optimization problems. They appear in many models in economics, game theory and mathematical physics. BL programs show a more complicated structure than standard finite problems. We study the so-called KKT-approach for solving bilevel problems, where the lower level minimality condition is replaced by the KKT- or the FJ-condition. This leads to a special structured mathematical program with complementarity constraints. We analyze the KKT-approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving BL problems numerically