1,928 research outputs found
On generalized semi-infinite optimization and bilevel optimization
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
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
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 convergence rate is
obtained locally.Comment: 30 pages, 7 figure
Convex Semi-Infinite programming: explicit optimality conditions
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
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