scholarly journals A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 908
Author(s):  
Md Sadikur Rahman ◽  
Ali Akbar Shaikh ◽  
Irfan Ali ◽  
Asoke Kumar Bhunia ◽  
Armin Fügenschuh
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Nonlinear Optimization ◽  
Optimization Problems ◽  
Interval Order ◽  
Constrained Optimization Problems ◽  
Interval Approach ◽  
Kkt Optimality Conditions ◽  
Interval Valued

In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty. Several researchers have discussed the interval approach to tackle nonlinear optimization uncertainty and derived the optimality conditions. However, there are several realistic situations in which the interval approach is not suitable. This study aims to introduce the Type-2 interval approach to overcome the limitation of the classical interval approach. This study introduces Type-2 interval order relation and Type-2 interval-valued function concepts to derive generalized KKT optimality conditions for constrained optimization problems under uncertain environments. Then, the optimality conditions are discussed for the unconstrained Type-2 interval-valued optimization problem and after that, using these conditions, generalized KKT conditions are derived. Finally, the proposed approach is demonstrated by numerical examples.

scholarly journals Sequential optimality conditions for cardinality-constrained optimization problems with applications

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Optimality Condition ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Local Minimizer ◽  
Constrained Optimization Problems ◽  
New Approach ◽  
Mathematical Programs ◽  
Sequential Optimality Conditions

AbstractRecently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.

scholarly journals Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems

2022 ◽  
Author(s):  
Helmut Gfrerer ◽  
Jane J. Ye ◽  
Jinchuan Zhou
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Support Function ◽  
Optimization Problems ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Second Order Optimality Conditions ◽  
Constrained Problems ◽  
Second Order Tangent Set ◽  
Order Tangent

In this paper, we study second-order optimality conditions for nonconvex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper, we propose two approaches for establishing second-order optimality conditions for the nonconvex case. In the first approach, we extend the concept of the support function so that it is applicable to general nonconvex set-constrained problems, whereas in the second approach, we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

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