Refinements of necessary optimality conditions in nondifferentiable programming I

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.

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Minimum Principle-Type Necessary Optimality Conditions in Scalar and Vector Optimization. An Account

Journal of Mathematics Research ◽

10.5539/jmr.v9n4p168 ◽

2017 ◽

Vol 9 (4) ◽

pp. 168

Author(s):

Giorgio Giorgi

Keyword(s):

Optimality Conditions ◽

Vector Optimization ◽

Optimization Problems ◽

Minimum Principle ◽

Necessary Optimality Conditions ◽

Equality Constraints ◽

Vector Optimization Problems ◽

Scalar Optimization ◽

Scalar And Vector Optimization

We take into condideration necessary optimality conditions of minimum principle-type, that is for optimization problems having, besides the usual inequality and/or equality constraints, a set constraint. The first part pf the paper is concerned with scalar optimization problems; the second part of the paper deals with vector optimization problems.

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Numerical Algorithms For State-Linear Optimal Impulsive Control Problems Based On Feedback Necessary Optimality Conditions

Cybernetics and Physics ◽

10.35470/2226-4116-2020-9-3-152-158 ◽

2020 ◽

pp. 152-158

Author(s):

Stepan Sorokin ◽

Maxim Staritsyn

Keyword(s):

Optimality Conditions ◽

Impulsive Control ◽

Numerical Algorithms ◽

Necessary Optimality Conditions ◽

Maximum Principles ◽

Control Problems ◽

Impulsive Systems ◽

Feedback Controls ◽

Non Local ◽

Optimal Impulsive Control

We propose and compare three numeric algorithms for optimal control of state-linear impulsive systems. The algorithms rely on the standard transformation of impulsive control problems through the discontinuous time rescaling, and the so-called “feedback”, direct and dual, maximum principles. The feedback maximum principles are variational necessary optimality conditions operating with feedback controls, which are designed through the usual constructions of the Pontryagin’s Maximum Principle (PMP); though these optimality conditions are formulated completely in the formalism of PMP, they essentially strengthen it. All the algorithms are non-local in the sense that they are aimed at improving non-optimal extrema of PMP (local minima), and, therefore, show the potential of global optimization.

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