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

A new regular multiplier embedding

summary:Embedding approaches can be used for solving non linear programs P. The idea is to define a one-parametric problem such that for some value of the parameter the corresponding problem is equivalent to P. A particular case is the multipliers embedding, where the solutions of the corresponding parametric problem can be interpreted as the points computed by the multipliers method on P. However, in the known cases, either path-following methods can not be applied or the necessary conditions for its convergence are fulfilled under very restrictive hypothesis. In this paper, we present a new multipliers embedding such that the objective function and the constraints of $P(t)$ are $C^3$ differentiable functions. We prove that the parametric problem satisfies the JJT-regularity generically, a necessary condition for the success of the path-following method

New embaddings for nonlinear multiobjective optimization problems I

In a dialogue procedure the decision maker has to determine in each step the aspiration and reservation level expressing his wishes (goals). This leads to an optimization problem wich is not easy to solve in the nonconvex case (the known starting point is not feasible). We propose a modified standard embedding (one parametric optimization). This problem will be discussed from the point of view of parametric optimization (non-degenerate critical points, singularities, pathfollowing methods to describe numericalley a connected component in the set of stationary points and in the set of generalized critical points, respectively, and jumps (descent methods) to other connected components in these sets). This embedding is much better for computing a goal realizer or replying that the goal was not realistic than the embeddings considered in the literature before, but in the worst case we have to find all connected components and this is an open problem

On the Role of the Mangasarian-Fromovitz Constraint Qualification for Penalty-, Exact Penalty- and Lagrange Multiplier Methods

In this paper we consider three embeddings (one-parametric optimization problems) motivated by penalty, exact penalty and Lagrange multiplier methods. We give an answer to the question under which conditions these methods are successful with an arbitrarily chosen starting point. Using the theory of one-parametric optimization (the local structure of the set of stationary points and of the set of generalized critical points, singularities, structural stability, pathfollowing and jumps) the so-called Mangasarian-Fromovitz condition and its extension play an important role. The analysis shows us that the class of optimization problems for which we can surely find a stationary point using a pathfollowing procedure for the modified penalty and exact penalty embedding is much larger than the class where the Lagrange multiplier embedding is successful. For the first class, the objective may be a “really non-convex” function, but for the second one we are restricted to convex optimization problems. This fact was a surprise at least for the authors

A modified standard embedding for linear complementarity problems

summary:We propose a modified standard embedding for solving the linear complementarity problem (LCP). This embedding is a special one-parametric optimization problem $P(t), t \in [0,1]$. Under the conditions (A3) (the Mangasarian–Fromovitz Constraint Qualification is satisfied for the feasible set $M(t)$ depending on the parameter $t$), (A4) ($P(t)$ is Jongen–Jonker– Twilt regular) and two technical assumptions, (A1) and (A2), there exists a path in the set of stationary points connecting the chosen starting point for $P(0)$ with a certain point for $P(1)$ and this point is a solution for the (LCP). This path may include types of singularities, namely points of Type 2 and Type 3 in the class of Jongen–Jonker–Twilt for $t\in [0,1)$. We can follow this path by using pathfollowing procedures (included in the program package PAFO). In case that the condition (A3) is not satisfied, also points of Type 4 and 5 may appear. The assumption (A4) will be justified by a perturbation theorem. Illustrative examples are presented