Stationary Point Optimality Conditions with Differentiability

1976 ◽  
pp. 114-132
Author(s):  
M. S. Bazaraa ◽  
C. M. Shetty
Keyword(s):  
Optimality Conditions ◽  
Stationary Point

scholarly journals Second- and First-Order Optimality Conditions in Vector Optimization

2015 ◽  
Vol 14 (04) ◽  
pp. 747-767 ◽  
Author(s):  
Vsevolod I. Ivanov
Keyword(s):  
Objective Function ◽  
Optimality Conditions ◽  
Stationary Point ◽  
Lagrange Function ◽  
Necessary Optimality Conditions ◽  
Second Order ◽  
Weak Efficiency ◽  
First Order ◽  
Vector Problem ◽  
First Order Optimality Conditions

In this paper, we obtain second- and first-order optimality conditions of Kuhn–Tucker type and Fritz John one for weak efficiency in the vector problem with inequality constraints. In the necessary conditions, we suppose that the objective function and the active constraints are continuously differentiable. We introduce notions of KTSP-invex problem and second-order KTSP-invex one. We obtain that the vector problem is (second-order) KTSP-invex if and only if for every triple [Formula: see text] with Lagrange multipliers [Formula: see text] and [Formula: see text] for the objective function and constraints, respectively, which satisfies the (second-order) necessary optimality conditions, the pair [Formula: see text] is a saddle point of the scalar Lagrange function with a fixed multiplier [Formula: see text]. We introduce notions second-order KT-pseudoinvex-I, second-order KT-pseudoinvex-II, second-order KT-invex problems. We prove that every second-order Kuhn–Tucker stationary point is a weak global Pareto minimizer (global Pareto minimizer) if and only if the problem is second-order KT-pseudoinvex-I (KT-pseudoinvex-II). It is derived that every second-order Kuhn–Tucker stationary point is a global solution of the weighting problem if and only if the vector problem is second-order KT-invex.

scholarly journals Optimality Conditions for Group Sparse Constrained Optimization Problems

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 84
Author(s):  
Wenying Wu ◽  
Dingtao Peng
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Stationary Point ◽  
Tangent Cone ◽  
Optimization Problems ◽  
Stationary Points ◽  
Tangent Cones ◽  
Sufficient Optimality Conditions ◽  
Normal Cones ◽  
The Relationship

In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Generalized Fixed-Structure Optimality Conditions for H2 Optimal Control

1993 ◽  
Author(s):  
Emmanuel G. Collins ◽  
Wassim M. Haddad ◽  
Sidney S. Ying
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Fixed Structure

Analysis and design of thermo-mechanical interfaces

2012 ◽  
Vol 60 (2) ◽  
pp. 205-213
Author(s):  
K. Dems ◽  
Z. Mróz
Keyword(s):  
Optimality Conditions ◽  
Mechanical Loading ◽  
Material Properties ◽  
Structural Response ◽  
External Boundary ◽  
Mechanical Loads ◽  
Internal Interface ◽  
Analysis And Design ◽  
First Order ◽  
Mechanical Nature

Abstract. An elastic structure subjected to thermal and mechanical loading with prescribed external boundary and varying internal interface is considered. The different thermal and mechanical nature of this interface is discussed, since the interface form and its properties affect strongly the structural response. The first-order sensitivities of an arbitrary thermal and mechanical behavioral functional with respect to shape and material properties of the interface are derived using the direct or adjoint approaches. Next the relevant optimality conditions are formulated. Some examples illustrate the applicability of proposed approach to control the structural response due to applied thermal and mechanical loads.

scholarly journals Robust Necessary Optimality Conditions for Nondifferentiable Complex Fractional Programming with Uncertain Data

2021 ◽  
Vol 189 (1) ◽  
pp. 221-243
Author(s):  
Jiawei Chen ◽  
Suliman Al-Homidan ◽  
Qamrul Hasan Ansari ◽  
Jun Li ◽  
Yibing Lv
Keyword(s):  
Optimality Conditions ◽  
Fractional Programming ◽  
Uncertain Data ◽  
Necessary Optimality Conditions

scholarly journals Topological Approach to Mathematical Programs with Switching Constraints

2021 ◽  
Author(s):  
Vladimir Shikhman
Keyword(s):  
Stationary Point ◽  
Mountain Pass Theorem ◽  
Strong Stability ◽  
Complete Characterization ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Linear Independence Constraint Qualification ◽  
Point Set ◽  
Stationary Level

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

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