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Mathematical programming with equilibrium constraints (MPEC) is the study of constrained optimization problems where the constraints include variational
Mathematical programming with equilibrium constraints
Mathematical_programming_with_equilibrium_constraints
problems modeled with EMP are reformulated to mathematical programs with equilibrium constraints (MPECs) and then they are solved with one of the GAMS
Extended Mathematical Programming
Extended_Mathematical_Programming
Method to solve optimization problems
Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique
Linear_programming
Study of mathematical algorithms for optimization problems
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria
Mathematical_optimization
Solution concept of a non-cooperative game
Self-confirming equilibrium - another relaxation of Nash equilibrium. Extended Mathematical Programming § Equilibrium Problems This term is dispreferred, as it can
Nash_equilibrium
Optimization software library
Mathematical programming with equilibrium constraints (MPEC). This version of IPOPT is generally known as IPOPT-C (with the 'C' standing for 'complementarity')
IPOPT
Topics referred to by the same term
to: Mathematical programming with equilibrium constraints Minor Planet Electronic Circular This disambiguation page lists articles associated with the
MPEC
Description of a system using mathematical concepts and language
mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social
Mathematical_model
Type of programming language
mathematical programs with equilibrium constraints constrained nonlinear systems general nonlinear problems non-linear programs with discontinuous derivatives
Algebraic_modeling_language
topological degree theory and nonlinear analysis. Mathematical programming with equilibrium constraints nl format for representing complementarity problems
Complementarity_theory
Mathematical optimization concept
a linear combination of the constraints, with positive coefficients, such that the coefficients of x in the constraints are at least cT. This linear
Dual_linear_program
Optimization problem in mathematics
In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and
Quadratically constrained quadratic program
Quadratically_constrained_quadratic_program
Physics of heat, work, and temperature
entropy, that increases as the constraints are removed, eventually reaching a maximum value at thermodynamic equilibrium, when the inhomogeneities practically
Thermodynamics
Quadratic fractional programming problem
optimization task and are commonly referred as mathematical programming problems with equilibrium constraints (MPEC). The upper level objective in such problems
Bilevel_optimization
of constraints over the variables is minimized. Distributed Constraint Satisfaction is a framework for describing a problem in terms of constraints that
Distributed constraint optimization
Distributed_constraint_optimization
Economic model
competition Extensive form game Industrial organization Mathematical programming with equilibrium constraints Leontief, Wassily (1936). "Stackelberg on Monopolistic
Stackelberg_competition
Situation where total gains match total losses
are most often solved with the minimax theorem which is closely related to linear programming duality, or with Nash equilibrium. In contrast, positive-sum
Zero-sum_game
libraries with significant optimization coverage. List of optimization software "The Nature of Mathematical Programming," Mathematical Programming Glossary
Comparison of optimization software
Comparison_of_optimization_software
Game theory solution
In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician
Correlated_equilibrium
Method to solve constrained optimization problems
extended to solve problems with multiple constraints using a similar argument. Consider a paraboloid subject to two line constraints that intersect at a single
Lagrange_multiplier
case of mathematical programming with equilibrium constraints (MPEC). Specifically, the likelihood function is maximized subject to the constraints imposed
Dynamic_discrete_choice
Topics referred to by the same term
Relaxation stands quite generally for a release of tension, a return to equilibrium. In the sciences, the term is used in the following ways: Relaxation
Relaxation
Class of economic models
general equilibrium) models. A CGE model consists of equations describing model variables and a database (usually very detailed) consistent with these model
Computable general equilibrium
Computable_general_equilibrium
Necessary condition for optimality associated with dynamic programming
maximizing utility, etc. The mathematical function that describes this objective is called the objective function. Dynamic programming breaks a multi-period
Bellman_equation
Theory of equilibrium between supply and demand
will result in an overall general equilibrium. General equilibrium theory contrasts with the theory of partial equilibrium, which analyzes a specific part
General_equilibrium_theory
Type of mathematical inequality
variational inequality Extended Mathematical Programming for Equilibrium Problems Mathematical programming with equilibrium constraints Obstacle problem Projected
Variational_inequality
Branch of applied mathematics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods
Mathematical_economics
Solution concept in game theory
notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is
Quantal_response_equilibrium
Chemical property
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical
Equilibrium_constant
Refinement of Nash equilibrium
Sequential equilibrium is a refinement of Nash equilibrium for extensive form games due to David M. Kreps and Robert Wilson. A sequential equilibrium specifies
Sequential_equilibrium
Standard example in game theory
identical. This makes the "both defect" case a weak equilibrium, compared with the strict equilibrium in the standard prisoner's dilemma. If a contestant
Prisoner's_dilemma
Behavior of individuals and firms
consumers may achieve equilibrium between preferences and expenditures by maximizing utility subject to consumer budget constraints. Production theory is
Microeconomics
Mathematical models of strategic interactions
the Nash equilibrium of the game in his Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles
Game_theory
Solution concept in game theory
Equilibrium (PBE) is a solution with Bayesian probability to a turn-based game with incomplete information. More specifically, it is an equilibrium concept
Perfect_Bayesian_equilibrium
Hungarian and American mathematician and physicist (1903–1957)
von Neumann read a book by the mathematical economist Léon Walras. Von Neumann noticed that Walras's General Equilibrium Theory and Walras's law, which
John_von_Neumann
When the ratio of reactants to products of a chemical reaction is constant with time
chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time
Chemical_equilibrium
Type of algorithm for constrained optimization
violation of the constraints. The measure of violation is nonzero when the constraints are violated and is zero in the region where constraints are not violated
Penalty_method
iterative partial least squares (NIPLS) Mathematical programming with equilibrium constraints — constraints include variational inequalities or complementarities
List of numerical analysis topics
List_of_numerical_analysis_topics
Type of perfect Bayesian equilibrium
In signaling games, a separating equilibrium is a type of perfect Bayesian equilibrium where agents with different characteristics choose different actions
Separating_equilibrium
Property of a thermodynamic system
spontaneous evolution cannot decrease with time. As a result, isolated systems evolve toward thermodynamic equilibrium, where the entropy is highest. "High"
Entropy
Specialist field of computer science
to show the epistemological constraints of computer-based simulation research. As computational science uses mathematical models representing the underlying
Computational_science
Class of games where players choose their actions sequentially
with temporal dynamics. Simultaneous game Subgame perfect equilibrium Sequential auction Brocas; Carrillo; Sachdeva (2018). "The Path to Equilibrium in
Sequential_game
sometimes called the "Nash conjecture," as it underlies the standard Nash equilibrium concept. However, alternative assumptions can be made. Suppose you have
Conjectural_variation
Economic model
interest in imperfect competition. Cournot presents a mathematical analysis of the equilibrium condition in a model of duopolist behaviour. This model
Cournot_competition
Aspect of game theory
In game theory, self-confirming equilibrium is a generalization of Nash equilibrium for extensive form games, in which players correctly predict the moves
Self-confirming_equilibrium
Macroeconomic method
Dynamic stochastic general equilibrium modeling (abbreviated as DSGE, or DGE, or sometimes SDGE) is a macroeconomic method which is often employed by
Dynamic stochastic general equilibrium
Dynamic_stochastic_general_equilibrium
Thought experiments
compares two different equilibrium states, after the process of adjustment (if any). It does not study the motion towards equilibrium, nor the process of
Comparative_statics
Concept in game theory
Coordination game Simultaneous game Surprisingly popular Equilibrium selection Rendezvous problem, the mathematical problem of maximising the probability of two people
Focal_point_(game_theory)
Use of mathematical and statistical methods in finance
Quantitative analysis in finance refers to the application of mathematical and statistical methods to problems in financial markets and investment management
Quantitative analysis (finance)
Quantitative_analysis_(finance)
Nash equilibrium of a bimatrix game algorithm
payoff is at most 1. The first m constraints require the probabilities to be non-negative, and the other n constraints require each of the n pure strategies
Lemke–Howson_algorithm
Game theory concept
theory, a subgame perfect equilibrium (SPE), or subgame perfect Nash equilibrium (SPNE), is a refinement of the Nash equilibrium concept, specifically designed
Subgame_perfect_equilibrium
keywords model, var, constraints. A model is an extension of a Java class that can contain not only fields and methods but also constraints and an objective
OptimJ
Solution concept in Game Theory
cursed equilibrium is a solution concept for static games of incomplete information. It is a generalization of the usual Bayesian Nash equilibrium, allowing
Cursed_equilibrium
Concept in game theory
Equilibrium selection is a concept from game theory which seeks to address reasons for players of a game to select a certain equilibrium over another.
Equilibrium_selection
Interdisciplinary field of research
Mathematical sociology is an interdisciplinary field of research concerned with the use of mathematics within sociological research. Starting in the early
Mathematical_sociology
game theory, a symmetric equilibrium is an equilibrium where all players use the same strategy (possibly mixed) in the equilibrium. In the Prisoner's Dilemma
Symmetric_equilibrium
Concept in game theory
In game theory, a strong Nash equilibrium (SNE) is a combination of actions of the different players, in which no coalition of players can cooperatively
Strong_Nash_equilibrium
Situation where players have only a small incentive to change strategies
epsilon-equilibrium, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no
Epsilon-equilibrium
Economic model of competition
as the Bertrand–Edgeworth model. With capacity constraints, there may not exist any pure strategy Nash equilibrium, the so-called Edgeworth paradox.
Bertrand_competition
Model of conflict for two players in game theory
are (D, C) and (C, D). There is also a mixed strategy equilibrium where each player Dares with probability 1/3. It results in expected payoffs of 14/3
Chicken_(game)
American economist
Journal of Mathematical Economics (2003), 39: 391–400. [CFDP 1360] "Monetary Equilibrium with Missing Markets" (with P. Dubey), Journal of Mathematical Economics
John_Geanakoplos
Solution Concept for Noncooperative games
In game theory, a satisfaction equilibrium is a solution concept for a class of non-cooperative games, namely games in satisfaction form. Games in satisfaction
Satisfaction_equilibrium
Hungarian mathematician and physicist (1847–1930)
finding the necessary condition for equilibrium is equivalent to minimizing the potential subject to constraints. This led to his formulation of the necessary
Gyula Farkas (natural scientist)
Gyula_Farkas_(natural_scientist)
Two-player coordination game in game theory
(1), in the mixed strategy equilibrium the man goes to the prize fight with probability 3/5 and the woman to the ballet with probability 3/5, so they end
Battle of the sexes (game theory)
Battle_of_the_sexes_(game_theory)
Concept in game theory
A Markov perfect equilibrium is an equilibrium concept in game theory. It has been used in analyses of industrial organization, macroeconomics, and political
Markov_perfect_equilibrium
Interdisciplinary research discipline
theoretical assumption of mathematical optimization by agents in equilibrium is replaced by the less restrictive postulate of agents with bounded rationality
Computational_economics
Economic Model
capacity constraint): he showed that if there is a fixed limit to what firms can sell, then there may exist no pure-strategy Nash equilibrium (this is
Bertrand–Edgeworth_model
1970s paradigm shift in economic thought, named for American economist Robert Lucas
"deep parameters" (relating to preferences, technology, and resource constraints) that are assumed to govern individual behavior: so-called "microfoundations"
Lucas_critique
Overview of finance and finance-related topics
optimization – Branch of numerical optimization Extended Mathematical Programming (§ EMP for stochastic programming) Genetic algorithm (List of genetic algorithm
Outline_of_finance
Game that repeats a base game
deviation. In other words, in the pricing competition game, the only Nash equilibrium is inefficient (for gas stations) that both charge p = c. This is more
Repeated_game
Simultaneous game found in game theory
and they both do better if they coordinate than if they played an off-equilibrium combination of actions. This setup can be extended to more than two strategies
Coordination_game
Solution concept in game theory
refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff
Risk_dominance
Transportation networks
paths once the system is in equilibrium. The user optimum equilibrium can be found by solving the following nonlinear programming problem min ∑ a ∫ 0 v a
Route_assignment
Equilibrium constants are determined in order to quantify chemical equilibria. When an equilibrium constant K is expressed as a concentration quotient
Determination of equilibrium constants
Determination_of_equilibrium_constants
Mathematical representation of economic system
relationships between them. The economic model is a simplified, often mathematical, framework designed to illustrate complex processes. Frequently, economic
Economic_model
Class of theorems about Nash equilibrium payoff profiles in repeated games
folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games (Friedman 1971). The original Folk
Folk_theorem_(game_theory)
the mathematical optimization problem is to minimize the internal energy dissipated along discontinuities, subject to nodal compatibility constraints. This
Discontinuity layout optimization
Discontinuity_layout_optimization
English saying meaning "equivalent retaliation"
disappear." Can be both Nash equilibrium and knife-edge equilibrium. Known as knife-edge equilibrium because the equilibrium "rests precariously on" the
Tit_for_tat
Decision rule used for minimizing the possible loss for a worst-case scenario
Nash equilibrium. In the context of zero-sum games, the minimax theorem is equivalent to:[failed verification] For every two-person zero-sum game with finitely
Minimax
Paradox in economics
describes a situation in which two players (firms) reach a state of Nash equilibrium where both firms charge a price equal to marginal cost ("MC"). The paradox
Bertrand_paradox_(economics)
Process of building computer models of energy systems in order to analyze them
research. Most rely on linear programming (including mixed-integer programming), although some use nonlinear programming. Solvers may use classical or
Energy_modeling
Complete plan on how a game player will behave in every possible game situation
the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2. During the 1980s, the concept
Strategy_(game_theory)
Logical paradox in decision-making theory
characterized as inherently socially disruptive, and are subject to legal constraints on their circulation as such, while the US has ruled that such materials
Paradox_of_tolerance
Set in game theory
game is nonempty if and only if the game is "balanced". Every Walrasian equilibrium has the core property, but not vice versa. The Edgeworth conjecture states
Core_(game_theory)
Israeli mathematician
probability theory, mathematical analysis, topology, set theory, and combinatorics, and has at times required the development of new mathematical tools within
Eilon_Solan
Solution concept for non-cooperative games
perfect equilibrium implements a weak version of backward induction, and increasingly stronger versions are sequential equilibrium, perfect equilibrium, quasi-perfect
Mertens-stable_equilibrium
Israeli-American mathematician (born 1930)
correlated equilibrium in game theory, which is a type of equilibrium in non-cooperative games that is more flexible than the classical Nash equilibrium. Furthermore
Robert_Aumann
Violations of the convexity assumptions of elementary economics
textbooks in microeconomics, general equilibrium theory, game theory, mathematical economics, and applied mathematics (for economists). The Shapley–Folkman
Non-convexity_(economics)
Work done by a force to move a particle along a virtual displacement
is in harmony with the given kinematic constraints." The argument is as follows. The principle of virtual work states that in equilibrium the virtual work
Virtual_work
Formulation of classical mechanics
applied to systems whose constraints, if any, are all holonomic. Three examples of nonholonomic constraints are: when the constraint equations are non-integrable
Lagrangian_mechanics
Solution concept in Game Theory
correlated equilibrium is a solution concept for static games of incomplete information. It is both a generalization of the correlated equilibrium perfect-information
Bayes_correlated_equilibrium
Significant topic in economics
textbooks in microeconomics, general equilibrium theory, game theory, mathematical economics, and applied mathematics (for economists). The Shapley–Folkman
Convexity_in_economics
Solution concept in game theory
Proper equilibrium in game theory is a refinement of Nash Equilibrium by Roger B. Myerson. Proper equilibrium further refines Reinhard Selten's notion
Proper_equilibrium
The concept of coalition-proof Nash equilibrium applies to certain "noncooperative" environments in which players can freely discuss their strategies
Coalition-proof Nash equilibrium
Coalition-proof_Nash_equilibrium
Solution concept capturing altruism in game theory
Berge equilibrium is a game theory solution concept named after the mathematician Claude Berge. It is similar to the standard Nash equilibrium, except
Berge_equilibrium
Mathematical model of the time dependence of a point in space
field of arithmetic geometry. Thermodynamic Formalism, The Mathematical Structures of Equilibrium Statistical Mechanics, Appendix A4.2, David Ruelle The measure
Dynamical_system
Overuse of a shared resource
organization Jevons paradox – Efficiency leads to increased demand Nash equilibrium – Solution concept of a non-cooperative game Overfishing – Removal of
Tragedy_of_the_commons
Concept in game theory
Theory with Applications to Economics. New York: Oxford University Press. pp. 209–215. ISBN 0-19-503660-3. "Shapley value", Encyclopedia of Mathematics, EMS
Shapley_value
traditional programs hold in the presence of constraints as well. Constraints play an important role in answer set programming because adding a constraint to a
Stable_model_semantics
Formal rule for predicting how a game will be played
game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium. Many solution concepts, for many games, will result
Solution_concept
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MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
MATHEMATICAL PROGRAMMING-WITH-EQUILIBRIUM-CONSTRAINTS
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