Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints

Movahedian, Nooshin; Nobakhtian, S.

doi:10.1016/j.na.2009.11.015

Cited by 28 publications

(7 citation statements)

References 18 publications

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“…These polynomials are formulated to exactly match the value of the derivatives when evaluated at the collocation points (τ i ). This relationship, assuming constant inputs over the time interval, is shown in Equation (2), where the derivatives at discrete time points are approximated as the summation of f evaluated at each collocation point (τ i ) multiplied by the corresponding interpolation polynomial (Ω j ). These additional equations allows the differential equation model to be solved as a nonlinear programming problem where differential terms are simply additional variables of an often large-scale and sparse system of nonlinear equations:…”

Section: Simultaneous Solution Methodsmentioning

confidence: 99%

“…Mathematical programs with equilibrium constraints (MPECs) have been proposed as a method to integrate non-smooth behavior into a set of simultaneous algebraic equations by the inclusion of complementarity conditions [2,3]. Complementarity, the requirement that at least one of a pair of variables be at some limit, provides a framework for representing disjunctive behavior using a set of continuous equations.…”

Section: Introductionmentioning

confidence: 99%

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A Continuous Formulation for Logical Decisions in Differential Algebraic Systems using Mathematical Programs with Complementarity Constraints

et al. 2016

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This work presents a methodology to represent logical decisions in differential algebraic equation simulation and constrained optimization problems using a set of continuous algebraic equations. The formulations may be used when state variables trigger a change in process dynamics, and introduces a pseudo-binary decision variable, which is continuous, but should only have valid solutions at values of either zero or one within a finite time horizon. This formulation enables dynamic optimization problems with logical disjunctions to be solved by simultaneous solution methods without using methods such as mixed integer programming. Several case studies are given to illustrate the value of this methodology including nonlinear model predictive control of a chemical reactor using a surge tank with overflow to buffer disturbances in feed flow rate. Although this work contains novel methodologies for solving dynamic algebraic equation (DAE) constrained problems where the system may experience an abrupt change in dynamics that may otherwise require a conditional statement, there remain substantial limitations to this methodology, including a limited domain where problems may converge and the possibility for ill-conditioning. Although the problems presented use only continuous algebraic equations, the formulation has inherent non-smoothness. Hence, these problems must be solved with care and only in select circumstances, such as in simulation or situations when the solution is expected to be near the solver's initial point.

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Section: Simultaneous Solution Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Continuous Formulation for Logical Decisions in Differential Algebraic Systems using Mathematical Programs with Complementarity Constraints

et al. 2016

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“…Mathematical programming problems with equilibrium constraints are applicable in many fields as hydroeconomic river-basin model [1], chemical engineering process [18], traffic and telecommunications networks [19], etc. For more details on mathematical programming problems with equilibrium constraints we refer to [2,3,4,5,6,8,12,15]. Pandey and Mishra [16] formulated Wolfe and Mond-Weir type dual models and established duality results for mathematical programming problems with equilibrium constraints.…”

Section: Introductionmentioning

confidence: 99%

Lagrange duality and saddle point optimality conditions for semi-infinite mathematical programming problems with equilibrium constraints

Singh

Maurya

Mishra

2019

YUGOSLAV J OPERATION

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In this paper, we consider a special class of optimization problems which contains infinitely many inequality constraints and finitely many complementarity constraints known as the semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC). We propose Lagrange type dual model for the SIMPEC and obtain their duality results using convexity assumptions. Further, we discuss the saddle point optimality conditions for the SIMPEC. Some examples are given to illustrate the obtained results.

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“…A mathematical program with equilibrium constraints (MPEC) usually refers to an optimization problem in which the essential constraints are defined by complementarity system or a parametric variational inequality. There are many equilibrium phenomena that arise from economics and engineering, characterized by either a variational inequality or an optimization problem, which justifies the name mathematical program with equilibrium constraints (MPEC) for the smooth case [35,10] and for the nonsmooth case [29,28,36]. Luo et al [20] presented a comprehensive study of MPEC.…”

Section: Introductionmentioning

confidence: 99%

On nonsmooth mathematical programs with equilibrium constraints using generalized convexity

Chandra

Mishra

Pankaj

2019

YUGOSLAV J OPERATION

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In this paper, we derive the sufficient condition for global optimality for a nonsmooth mathematical program with equilibrium constraints involving generalized invexity. We formulate the Wolfe and Mond-Weir type dual models for the problem using convexificators. We establish weak and strong duality theorems to relate the mathematical program with equilibrium constraints and the dual models in the framework of convexificators.

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Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints

Cited by 28 publications

References 18 publications

A Continuous Formulation for Logical Decisions in Differential Algebraic Systems using Mathematical Programs with Complementarity Constraints

A Continuous Formulation for Logical Decisions in Differential Algebraic Systems using Mathematical Programs with Complementarity Constraints

Lagrange duality and saddle point optimality conditions for semi-infinite mathematical programming problems with equilibrium constraints

On nonsmooth mathematical programs with equilibrium constraints using generalized convexity

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