scholarly journals CONSTRAINT QUALIFICATIONS IN PARTIAL IDENTIFICATION

2021 ◽  
pp. 1-24
Author(s):  
Hiroaki Kaido ◽  
Francesca Molinari ◽  
Jörg Stoye
Keyword(s):  
Stochastic Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Partial Identification ◽  
Moment Inequalities ◽  
Pure Moment ◽  
High Level

The literature on stochastic programming typically restricts attention to problems that fulfill constraint qualifications. The literature on estimation and inference under partial identification frequently restricts the geometry of identified sets with diverse high-level assumptions. These superficially appear to be different approaches to closely related problems. We extensively analyze their relation. Among other things, we show that for partial identification through pure moment inequalities, numerous assumptions from the literature essentially coincide with the Mangasarian–Fromowitz constraint qualification. This clarifies the relation between well-known contributions, including within econometrics, and elucidates stringency, as well as ease of verification, of some high-level assumptions in seminal papers.

scholarly journals Exact Penalization in Stochastic Programming—Calmness and Constraint Qualification

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Martin Branda
Keyword(s):  
Stochastic Programming ◽  
Error Bound ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Discrete Distributions ◽  
Exact Penalty ◽  
Asymptotic Equivalence ◽  
Exact Penalization ◽  
Local Minimizers ◽  
Constrained Problems

We deal with the conditions which ensure exact penalization in stochastic programming problems under finite discrete distributions. We give several sufficient conditions for problem calmness including graph calmness, existence of an error bound, and generalized Mangasarian-Fromowitz constraint qualification. We propose a new version of the theorem on asymptotic equivalence of local minimizers of chance constrained problems and problems with exact penalty objective. We apply the theory to a problem with a stochastic vanishing constraint.

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

2018 ◽  
Vol 35 (01) ◽  
pp. 1850008
Author(s):  
Na Xu ◽  
Xide Zhu ◽  
Li-Ping Pang ◽  
Jian Lv
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Accumulation Point ◽  
Stationary Points ◽  
Nondegeneracy Condition ◽  
Equilibrium Constraints ◽  
Convergence Properties ◽  
Constant Rank Constraint Qualification ◽  
Relaxation Schemes

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

scholarly journals A comparison of constraint qualifications in infinite-dimensional convex programming revisited

1999 ◽  
Vol 40 (3) ◽  
pp. 353-378 ◽  
Author(s):  
C. Zặlinescu
Keyword(s):  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Strong Duality ◽  
Sufficient Condition ◽  
Subdifferential Calculus ◽  
Infinite Dimensional ◽  
Qualification Conditions ◽  
The One

In 1990 Gowda and Teboulle published the paper [16], making a comparison of several conditions ensuring the Fenchel-Rockafellar duality formulainf{f(x) + g(Ax) | x ∈ X} = max{−f*(A*y*) − g*(− y*) | y* ∈ Y*}.Probably the first comparison of different constraint qualification conditions was made by Hiriart-Urruty [17] in connection with ε-subdifferential calculus. Among them appears, as the basic sufficient condition, the formula for the conjugate of the corresponding function; such functions are: f1 + f2, g o A, max{fl,…, fn}, etc. In fact strong duality formulae (like the one above) and good formulae for conjugates are equivalent and they can be used to obtain formulae for ε-subdifferentials, using a technique developed in [17] and extensively used in [46].

scholarly journals A Review of Stochastic Programming Methods for Optimization of Process Systems Under Uncertainty

2021 ◽  
Vol 2 ◽  
Author(s):  
Can Li ◽  
Ignacio E. Grossmann
Keyword(s):  
Stochastic Programming ◽  
Systems Engineering ◽  
State Of The Art ◽  
Mathematical Framework ◽  
Process Systems Engineering ◽  
Process Industries ◽  
Process Systems ◽  
Current State ◽  
Engineering Problems ◽  
High Level

Uncertainties are widespread in the optimization of process systems, such as uncertainties in process technologies, prices, and customer demands. In this paper, we review the basic concepts and recent advances of a risk-neutral mathematical framework called “stochastic programming” and its applications in solving process systems engineering problems under uncertainty. This review intends to provide both a tutorial for beginners without prior experience and a high-level overview of the current state-of-the-art developments for experts in process systems engineering and stochastic programming. The mathematical formulations and algorithms for two-stage and multistage stochastic programming are reviewed with illustrative examples from process industries. The differences between stochastic programming under exogenous uncertainty and endogenous uncertainties are discussed. The concepts and several data-driven methods for generating scenario trees are also reviewed.

scholarly journals On an exact penality result and new constraint qualifications for mathematical programs with vanishing constraints

2019 ◽  
Vol 29 (3) ◽  
pp. 309-324
Author(s):  
Triloki Nath ◽  
Abeka Khare
Keyword(s):  
Penalty Function ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Mathematical Programs ◽  
Vanishing Constraints ◽  
Abadie Constraint Qualification

In this paper, we considered the mathematical programs with vanishing constraints or MPVC. We proved that an MPVC-tailored penalty function, introduced in [5], is still exact under a very weak and new constraint qualification. Most importantly, this constraint qualification is shown to be strictly stronger than MPVC-Abadie constraint qualification.

Moment Inequalities and Partial Identification in Industrial Organization

2021 ◽  
Author(s):  
Brendan Kline ◽  
Ariel Pakes ◽  
Elie Tamer
Keyword(s):  
Industrial Organization ◽  
Partial Identification ◽  
Moment Inequalities

New Constraint Qualifications for S-Stationarity for MPEC with Nonsmooth Objective

2019 ◽  
Vol 36 (02) ◽  
pp. 1940001
Author(s):  
Peng Zhang ◽  
Jin Zhang ◽  
Gui-Hua Lin ◽  
Xinmin Yang
Keyword(s):  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Mathematical Problem ◽  
Equilibrium Constraints ◽  
Lipschitz Continuous ◽  
Nonlinear Programs ◽  
Locally Lipschitz ◽  
Locally Lipschitz Continuous

This paper considers a mathematical problem with equilibrium constraints (MPEC) in which the objective is locally Lipschitz continuous but not continuously differentiable everywhere. Our focus is on constraint qualifications for the nonsmooth S-stationarity in the sense of the limiting subdifferentials. First, although the MPEC-LICQ is not a constraint qualification for the nonsmooth S-stationarity, we show that the MPEC-LICQ can serve as a constraint qualification for the nonsmooth S-stationarity under some kind of regularity. Then, we extend some new constraint qualifications for nonlinear programs to the considered nonsmooth MPEC and show that all of them can serve as constraint qualifications for the nonsmooth S-stationarity. We further extend these results to the multiobjective case.

scholarly journals Duality theorems for convex programming without constraint qualification

1984 ◽  
Vol 36 (2) ◽  
pp. 253-266 ◽  
Author(s):  
P. Kanniappan
Keyword(s):  
Convex Function ◽  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Convex Programming Problem ◽  
Duality Theorems ◽  
Finite Dimensional ◽  
Lower Semi ◽  
Continuous Convex Function

AbstractInvoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.

scholarly journals Asymptotic stationarity and regularity for nonsmooth optimization problems

2020 ◽  
Author(s):  
Patrick Mehlitz
Keyword(s):  
Nonsmooth Optimization ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Variational Calculus ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Disjunctive Constraints ◽  
The Relationship ◽  
Type Constraint

Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this new sense are already Mordukhovich-stationary, the presence of a constraint qualification which we call AM-regularity is necessary. We investigate the relationship between AM-regularity and other constraint qualifications from nonsmooth optimization like metric (sub-)regularity of the underlying feasibility mapping. Our findings are applied to optimization problems with geometric and, particularly, disjunctive constraints. This way, it is shown that AM-regularity recovers recently introduced cone-continuity-type constraint qualifications, sometimes referred to as AKKT-regularity, from standard nonlinear and complementarity-constrained optimization. Finally, we discuss some consequences of AM-regularity for the limiting variational calculus.

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.

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