1,928 research outputs found

    On generalized semi-infinite optimization and bilevel optimization

    Get PDF
    The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL). We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems

    Linear convergence of accelerated conditional gradient algorithms in spaces of measures

    Full text link
    A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear O(1/k)\mathcal{O}(1/k) rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear O(ζk)\mathcal{O}(\zeta^k) convergence rate is obtained locally.Comment: 30 pages, 7 figure

    Convex Semi-Infinite programming: explicit optimality conditions

    Get PDF
    We consider the convex Semi-In¯nite Programming (SIP) problem where objec- tive function and constraint function are convex w.r.t. a ¯nite-dimensional variable x and all of these functions are su±ciently smooth in their domains. The constraint function depends also on so called time variable t that is de¯ned on the compact set T ½ R. In our recent paper [15] the new concept of immobility order of the points of the set T was introduced and the Implicit Optimality Criterion was proved for the convex SIP problem under consideration. In this paper the Implicit Optimality Criterion is used to obtain new ¯rst and second order explicit optimality conditions. We consider separately problems that satisfy and that do not satisfy the the Slater condition. In the case of SIP problems with linear w.r.t. x constraints the optimal- ity conditions have a form of the criterion. Comparison of the results obtained with some other known optimality conditions for SIP problems is provided as well
    corecore