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 An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

scholarly journals Strong complementary approximate Karush-Kuhn-Tucker conditions for multiobjective optimization problems

2021 ◽  
pp. 24-24
Author(s):  
Jitendra Maurya ◽  
Shashi Mishra
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Inequality Constraints ◽  
Multiobjective Optimization Problems ◽  
Sequential Optimality Conditions ◽  
Equality And Inequality Constraints ◽  
Optim Theory

In this paper, we establish strong complementary approximate Karush- Kuhn-Tucker (SCAKKT) sequential optimality conditions for multiobjective optimization problems with equality and inequality constraints without any constraint qualifications and introduce a weak constraint qualification which assures the equivalence between SCAKKT and the strong Karush-Kuhn-Tucker (J Optim Theory Appl 80 (3): 483{500, 1994) conditions for multiobjective optimization problems.

The Rate of Convergence of a NLM Based on F–B NCP for Constrained Optimization Problems Without Strict Complementarity

2015 ◽  
Vol 32 (03) ◽  
pp. 1550012 ◽  
Author(s):  
Suxiang He ◽  
Liwei Zhang ◽  
Jie Zhang
Keyword(s):  
Constrained Optimization ◽  
Rate Of Convergence ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Inequality Constraints ◽  
Constrained Optimization Problems ◽  
Strict Complementarity ◽  
Linear Rate ◽  
Nonlinear Lagrangian

It is well-known that the linear rate of convergence can be established for the classical augmented Lagrangian method for constrained optimization problems without strict complementarity. Whether this result is still valid for other nonlinear Lagrangian methods (NLM) is an interesting problem. This paper proposes a nonlinear Lagrangian function based on Fischer–Burmeister (F–B) nonlinear complimentarity problem (NCP) function for constrained optimization problems. The rate of convergence of this NLM is analyzed under the linear independent constraint qualification and the strong second-order sufficient condition without strict complementarity when subproblems are assumed to be solved exactly and inexactly, respectively. Interestingly, it is demonstrated that the Lagrange multipliers associating with inactive inequality constraints at the local minimum point converge to zeros superlinearly. Several illustrative examples are reported to show the behavior of the NLM.

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.

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