Related papers: On Farkas' Lemma and Related Propositions in BISH
Farkas' Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of…
Given $A\in \Z^{m\times n}$ and $b\in\Z^m$, we consider the issue of existence of a nonnegative integral solution $x\in \N^n$ to the system of linear equations $Ax=b$. We provide a discrete and explicit analogue of the celebrated Farkas…
Farkas' lemma is an ubiquitous tool in optimisation, as it provides necessary and sufficient conditions to have $b \in A(P)$, where $P$ is a closed convex cone, $A$ is a (continuous) linear mapping and $b$ is a fixed vector. The standard…
Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over…
We prove the almost equivalence of the minimax theorem and the strong duality theorem for a large class of games and conic programs. The previous fundamental results on the equivalence of linear programming and two-player zero-sum games…
Using Sperner's lemma for modified partition of a simplex we will constructively prove the existence of a Nash equilibrium in a finite strategic game with sequentially locally non-constant payoff functions. We follow the Bishop style…
We unify nonlinear Farkas lemma and S-lemma to a generalized alternative theorem for nonlinear nonconvex system. It provides fruitful applications in globally solving nonconvex non-quadratic optimization problems via revealing the hidden…
A new version of Farkas lemma of alternative linear systems is proposed. One and the same matrix $A$ and vector $b$ have always been used in alternative linear systems. The paper shows a different way of alternative systems involving…
This paper generalizes the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) lemma to the case of weak topology, and obtains the Ky Fan minimax inequality defined on non-empty non-compact convex subsets in reflexive Banach spaces, then we apply it…
We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we…
Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas' lemma, but…
Our first result is a statement of a somewhat general form of a non-substitution theorem for linear programming problems, along with a very easy proof of the same. Subsequently, we provide an easy proof of theorem 1 in a 1979 paper of Olvi…
In the framework of stochastic zero-sum differential games, we establish a verification theorem, inspired by those existing in stochastic control, to provide sufficient conditions for a pair of feedback controls to form a Nash equilibrium.…
The recent theory of sequential games and selection functions by Mar- tin Escardo and Paulo Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is…
We study constructively the relations between the finite cases of Dickson's lemma. Although there are many constructive proofs of them, the novel aspect of our proofs is the extraction of a corresponding bound. We provide some new one-step…
The first two authors of this paper asserted in Lemma 4 of "New Farkas-type constraint qualifications in convex infinite programming" (DOI: 10.1051/cocv:2007027) that a given reverse convex inequality is consequence of a given convex system…
The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig (1951) is known to be incomplete. We explain and combine classical…
Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear…
In this paper we are concerned with a Gordan-type theorem involving an arbitrary number of inequality functions. We not only state its validity under a weak convexity assumption on the functions, but also show it is an optimal result. We…
Exploiting the algebraic structure of the set of bimatrix games, a divide-and-conquer algorithm for finding Nash equilibria is proposed. The algorithm is fixed-parameter tractable with the size of the largest irreducible component of a game…