253 research outputs found
From the Farkas Lemma to the Hahn–Banach Theorem
This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ◦ g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn–Banach–Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn–Banach theorem and the Mazur–Orlicz theorem for extended sublinear functions.This research was partially supported by MINECO of Spain, grant MTM2011-29064-C03-02, and by the NAFOSTED of Vietnam
Regularity for evolution equations with non-autonomous perturbations in Banach spaces
We provide regularity of solutions to a large class of evolution equations on
Banach spaces where the generator is composed of a static principal part plus a
non-autonomous perturbation. Regularity is examined with respect to the graph
norm of the iterations of the principal part. The results are applied to the
Schr\"odinger equation and conditions on a time-dependent scalar potential for
regularity of the solution in higher Sobolev spaces are derived
A primal--dual algorithm as applied to optimal control problems
We propose a primal--dual technique that applies to infinite dimensional
equality constrained problems, in particular those arising from optimal
control. As an application of our general framework, we solve a
control-constrained double integrator optimal control problem and the
challenging control-constrained free flying robot optimal control problem by
means of our primal--dual scheme. The algorithm we use is an
epsilon-subgradient method that can also be interpreted as a penalty function
method. We provide extensive comparisons of our approach with a traditional
numerical approach
A convex analysis approach to optimal controls with switching structure for partial differential equations
Optimal control problems involving hybrid binary-continuous control costs are
challenging due to their lack of convexity and weak lower semicontinuity.
Replacing such costs with their convex relaxation leads to a primal-dual
optimality system that allows an explicit pointwise characterization and whose
Moreau-Yosida regularization is amenable to a semismooth Newton method in
function space. This approach is especially suited for computing switching
controls for partial differential equations. In this case, the optimality gap
between the original functional and its relaxation can be estimated and shown
to be zero for controls with switching structure. Numerical examples illustrate
the effectiveness of this approach
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