Book ChapterDOI
Parametric optimization: Pathfollowing with jumps
Jürgen Guddat,Hubertus Th. Jongen,Dieter Nowack +2 more
- pp 43-53
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In this article, a partial concept for finding a suitably fine discretization for finite dimensional optimization problems depending on one real parameter t was proposed, which is based on the generic behavior of such problems.Abstract:
We consider finite dimensional optimization problems depending on one real parameter t Recently, Jongen/Jonker/Twilt [9] studied the generic behaviour of such problems Based on this investigation, we propose a partial concept for finding a suitably fine discretization 0=to<…<ti−1<ti<…<tN=1 of the interval [0,1], and corresponding local minima x(ti), i=1,…,N; here, information on the point x(ti-1) is used in order to compute x(ti) Mainly, socalled continuation methods can be exploited However, at some parameter values, the branch of local minima used might have an endpoint; at such points one has to jump to another branch of local minima in order to continue the execution of the desired process In case that the feasible set in a neighborhood of such a mentioned endpoint remains nonempty for increasing parameter values, it will be shown how a jump can be realizedread more
Citations
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A Complete Bibliography of Lecture Notes in Mathematics (1985{1989)
TL;DR: (1 < p ≤ ∞) [LS87f] (2) [HR88a].
Journal ArticleDOI
On parametric nonlinear programming
TL;DR: In this paper, a tutorial survey of finite dimensional optimization problems which depend on parameters is presented, focusing on unfolding and singularity theory, structural analysis of families of constraint sets, constrained optimization problems and semi-infinite optimization.
Journal ArticleDOI
An implicit-function theorem for C0, 1-equations and parametric C1, 1-optimization
TL;DR: The implicit function theorem deals with the solutions of the equation F (x, t ) = a for locally Lipschitz functions F from R n + m into R n, where the existence of a locally well-defined and Linschitzian solution function x = G (a, t ) will be completely characterized in terms of certain multivalued directional derivatives of F which determine the corresponding derivatives of G in a simple way as discussed by the authors.
Journal ArticleDOI
On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach
TL;DR: This paper proposes a rigorous active set management strategy on top of a continuation method based on interval analysis, certified with respect to feasibility, local optimality and connectivity, which allows overcoming the latter limitation as illustrated on a representative bi-objective problem.
Journal ArticleDOI
A multi-parametric recursive continuation method for nonlinear dynamical systems
TL;DR: In this paper, an efficient multi-parametric recursive continuation method of specific solution points of a nonlinear dynamical system such as bifurcation points is proposed, which explores the topology of specific points found on the frequency response curves by tracking extremum points in the successive codimensions of the problem with respect to multiple system parameters.
References
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Book
Optimization-Theory and Applications
TL;DR: Theoretical Equivalence of Mayer, Lagrange, and Bolza Problems of Optimal Control, and the Necessary Conditions and Sufficient Conditions Convexity and Lower Semicontinuity.
Journal ArticleDOI
On the completeness and constructiveness of parametric characterizations to vector optimization problems
TL;DR: In this paper, the authors present a methodological approach to compare characterizations of optimal solutions to vector optimization problems and by applications to decision support systems, and present an impossibility theorem of complete and robustly computable characterization of efficient (as opposed to weakly or properly efficient) solutions.
Journal ArticleDOI
Critical sets in parametric optimization
H. Th. Jongen,P. Jonker,F. Twilt +2 more
TL;DR: In this paper, the concept of generalized critical points (g.c. point) is introduced and the set of points of ∑ can be divided into five (characteristic) types: non-degenerate critical points, strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints.