On quasi approximate solutions for nonsmooth robust semi-infinite optimization problems

2019 ◽  
Vol 35 (3) ◽  
pp. 417-426 ◽  
Author(s):  
CHANOKSUDA KHANTREE ◽  
RABIAN WANGKEEREE ◽  
◽  
Keyword(s):  
Constraint Qualification ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Data Uncertainty ◽  
Necessary Condition ◽  
Duality Theorems ◽  
Approximate Form ◽  
Locally Lipschitz ◽  
Approximate Duality

This paper devotes to the quasi ε-solution for robust semi-infinite optimization problems (RSIP) involving a locally Lipschitz objective function and infinitely many locally Lipschitz constraint functions with data uncertainty. Under the fulfillment of robust type Guignard constraint qualification and robust type Kuhn-Tucker constraint qualification, a necessary condition for a quasi ε-solution to problem (RSIP). After introducing the generalized convexity, we give a sufficient optimality for such a quasi ε-solution to problem (RSIP). Finally, we also establish approximate duality theorems in term of Wolfe type which is formulated in approximate form.

scholarly journals On approximate solutions for convex semi-infinite programming with uncertainty

2020 ◽  
Author(s):  
Ke Su ◽  
Yumeng Lin ◽  
Chen Wang
Keyword(s):  
Approximate Solution ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Necessary Condition ◽  
Optimization Approach ◽  
Duality Results ◽  
Cone Constraint ◽  
Lagrangian Dual ◽  
Infinite Programming

In this paper, we consider approximate solutions (also called $\varepsilon$-solutions) for semi-infinite optimization problems that objective function and constraint functions with uncertainty data are all convex, and establish robust counterpart of convex semi-infinite program and then consider approximate solutions for its. Moreover, the robust necessary condition and robust sufficient theorems are obtained. Then the duality results of the Lagrangian dual approximate solution is given by the robust optimization approach under a cone constraint qualification.

scholarly journals An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems

2021 ◽  
Author(s):  
Bennet Gebken ◽  
Sebastian Peitz
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Real Life ◽  
Practical Method ◽  
Descent Method ◽  
Bundle Method ◽  
Necessary Condition ◽  
Test Problems ◽  
Multiobjective Optimization Problems ◽  
Locally Lipschitz

AbstractWe present an efficient descent method for unconstrained, locally Lipschitz multiobjective optimization problems. The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth multiobjective optimization problems with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method with the multiobjective proximal bundle method by Mäkelä. The results indicate that our method is competitive while being easier to implement. Although the number of objective function evaluations is larger, the overall number of subgradient evaluations is smaller. Our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth problems. Finally, we demonstrate how our method can be used for solving sparse optimization problems, which are present in many real-life applications.

scholarly journals On optimality conditions for robust weak sharp solution in uncertain optimizations

2020 ◽  
Vol 36 (3) ◽  
pp. 443-452
Author(s):  
JUTAMAS KERDKAEW ◽  
RABIAN WANGKEEREE ◽  
GUE MYUNG LEE
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Generalized Convexity ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Robust Version ◽  
Robust Optimization Problem ◽  
Weak Sharp Solution ◽  
Weak Sharp

In this paper, we investigate the robust optimization problem involving nonsmooth and nonconvex real-valued functions. We firstly establish a necessary condition for the local robust weak sharp solution of considered problem under a constraint qualification. These optimality conditions are presented in terms of multipliers and Mordukhovich subdifferentials of the related functions. Then, by employing the robust version of the (KKT) condition, and some appropriate generalized convexity conditions, we also obtain some sufficient conditions for the global robust weak sharp solutions of the problem. In addition, some examples are presented for illustrating or supporting the results.

scholarly journals Duality theorems for nondifferentiable semi-infinite interval-valued optimization problems with vanishing constraints

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Haijun Wang ◽  
Huihui Wang
Keyword(s):  
Optimization Problem ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Strong Duality ◽  
Infinite Interval ◽  
Duality Theorems ◽  
Converse Duality ◽  
Vanishing Constraints ◽  
Dual Models ◽  
Interval Valued

AbstractIn this paper, we study the duality theorems of a nondifferentiable semi-infinite interval-valued optimization problem with vanishing constraints (IOPVC). By constructing the Wolfe and Mond–Weir type dual models, we give the weak duality, strong duality, converse duality, restricted converse duality, and strict converse duality theorems between IOPVC and its corresponding dual models under the assumptions of generalized convexity.

ϵ-Efficient solutions in semi-infinite multiobjective optimization

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1397-1410
Author(s):  
Tatiana Shitkovskaya ◽  
Do Sang Kim
Keyword(s):  
Multiobjective Optimization ◽  
Nonsmooth Analysis ◽  
Constraint Qualification ◽  
Dual Problem ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Duality Theorems ◽  
Multiobjective Optimization Problems ◽  
Special Cases ◽  
Scalarization Method

In this paper we apply some tools of nonsmooth analysis and scalarization method due to Chankong–Haimes to find ϵ-efficient solutions of semi-infinite multiobjective optimization problems (MP). We establish ϵ-optimality conditions of Karush–Kuhn–Tucker (KKT) type under Farkas–Minkowski (FM) constraint qualification by using ϵ-subdifferential concept. In addition we propose mixed type dual problem (including dual problems of Wolfe and Mond–Weir types as special cases) for ϵ-efficient solutions and investigate relationship between mentioned (MP) and its dual problem as well as establish several ϵ-duality theorems.

scholarly journals Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators

2022 ◽  
Vol 12 (1) ◽  
pp. 93
Author(s):  
Jutamas Kerdkaew ◽  
Rabian Wangkeeree ◽  
Rattanaporn Wangkeeree
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Data Uncertainty ◽  
Global Optimality ◽  
Optimal Solutions ◽  
Duality Theorems ◽  
Uncertain Optimization ◽  
Kkt Point ◽  
Robust Duality

<p style='text-indent:20px;'>In this paper, a robust optimization problem, which features a maximum function of continuously differentiable functions as its objective function, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisfies the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were first introduced by Jayakumar and Srisatkunarajah [<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.</p>

scholarly journals Higher order duality of multiobjective constrained ratio optimization problems

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1985-1998
Author(s):  
Arshpreet Kaur ◽  
Navdeep Kailey ◽  
M.K. Sharma
Keyword(s):  
Generalized Convexity ◽  
Feasible Solution ◽  
Optimization Problems ◽  
Higher Order ◽  
Type I ◽  
Dual Model ◽  
Duality Theorems ◽  
Fractional Programs ◽  
Ratio Optimization ◽  
Higher Order Duality

A new concept in generalized convexity, called higher order (C,?,?,?,d) type-I functions, is introduced. To show the existence of such type of functions, we identify a function lying exclusively in the class of higher order (C,?,?,?,d) type-I functions and not in the class of (C,?,?,?,d) type-I functions already existing in the literature. Based upon the higher order (C,?,?,?,d) type-I functions, the optimality conditions for a feasible solution to be an efficient solution are derived. A higher order Schaible dual has been then formulated for nondifferentiable multiobjective fractional programs. Weak, strong and strict converse duality theorems are established for higher order Schaible dual model and relevant proofs are given under the aforesaid function.

scholarly journals R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Bilevel Optimization ◽  
Regularity Property ◽  
Constant Rank Constraint Qualification ◽  
Existence Criterion ◽  
Set Valued Mappings

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

scholarly journals Duality Theorems for (ρ,ψ,d)-Quasiinvex Multiobjective Optimization Problems with Interval-Valued Components

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 894
Author(s):  
Savin Treanţă
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Integral Functional ◽  
Multiple Integral ◽  
Control Problems ◽  
Duality Theorems ◽  
New Class ◽  
Duality Results ◽  
Multiobjective Optimization Problems ◽  
Interval Valued

The present paper deals with a duality study associated with a new class of multiobjective optimization problems that include the interval-valued components of the ratio vector. More precisely, by using the new notion of (ρ,ψ,d)-quasiinvexity associated with an interval-valued multiple-integral functional, we formulate and prove weak, strong, and converse duality results for the considered class of variational control problems.

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.

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