28,237 research outputs found

    Variational Analysis And Stability In Optimization

    Get PDF
    The dissertation is devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from both viewpoints of optimization theory and its applications. Employing second-order subdifferentials of variational analysis, we obtain necessary and sufficient conditions for fully stable local minimizers in general classes of constrained optimization problems including problems of composite optimization as well as problems of nonlinear programming with twice continuously differentiable data. Based on our recent explicit calculations of the second-order subdifferential for convex piecewise linear functions, we establish relationships between nondegeneracy and second-order qualification for fully amenable compositions involving piecewise linear functions and obtain new applications of the developed second-order theory to full stability in composite optimization and constrained minimax problems, strong regularity of associate generalized equations and strong stability of stationary points for composite optimization. Finally, we discuss the important concept of critical multipliers for composite optimization problems and characterize it via second-order subdifferentials. Then we demonstrate that full stability can rule out the existence of critical multipliers in the mentioned framework

    Second-order subdifferential calculus with applications to tilt stability in optimization

    Get PDF
    The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms

    Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis

    Full text link
    We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale (Dynamical systems, New York, 1973, pp. 531-544). The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set and stable Pareto critical set, and can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure
    • …
    corecore