Farkas’ lemma: three decades of generalizations for mathematical optimization

In this paper we present a survey of generalizations of the celebrated Farkas’s lemma, starting from systems of linear inequalities to a broad variety of non-linear systems. We focus on the generalizations which are targeted towards applications in continuous optimization. We also briefly describe the main applications of generalized Farkas’ lemmas to continuous optimization problems.     

Zaks’ Lemma for Coherent Rings

Let A be a left and right coherent ring and C _A (resp., $C_{A^{\mathrm{op}}}$ ) a minimal cogenerator for right (resp., left) A-modules. We show that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}}$ whenever flat dim C _A ?<?∞ and $\mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ , and that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ if and only if the finitely presented right A-modules have bounded Gorenstein dimension.     

A Network Algorithm for Realizing Constructivelythe Farkas Lemma Arising from the Quasi-Generalized Network

1.IntroductionLetRIbethespaceofallrealvectorsindexedbyelementsofafiniteseti,icachosenelemelltofi,SandS*apairofcompletelyorthogonalsubspacesofRI.Foragivenpartition(PI,P2,P3,P4)ofI--{ic}(i.e.PIuPZuP3uP4=I--{eo},andPinPj=acfori/j),letF={axled6S,Ax'.>0,Ax.20fore6PI,Ax.50foreEP2,Ax.~0foreEP3},F*={dyIAyES*,ac.>0,ace30foreEPI,ac50foreEP2,ac.~0foreEP4}.Now,theFarkasLemmacanbegenerallydescribedas[1]:oneandonlyoneofthefollowingtwostatemelltsholds:(i)ThereexistsaaxEF.(n)Thereexistsaac…     

A Note on S-Nakayama’s Lemma

Ukrainian Mathematical Journal - We propose an S-version of Nakayama’s lemma. Let R be a commutative ring, let S be a multiplicative subset of R, and let M be an S-finite R-module. Also let I...     

Several generalizations of Tverberg’s theorem

In a generalization of Radon’s theorem, Tverberg showed that each setS of at least (d+1) (r ? 1)+1 points inR ^d has anr-partition into (pair wise disjoint) subsetsS =S ₁ ∪ … ∪S _r so that \(\bigcap\nolimits_i^r {\underline{\underline {}} } _1 \) convS _i # Ø. This note considers the following more general problems: (1) How large mustS σR ^d be to assure thatS has anr-partitionS=S ₁∪ … ∪S _r so that eachn members of the family {convS _i ～ _i-1 ^r have non-empty intersection, where 1<=n<=r. (2) How large mustS ∪R ^d be to assure thatS has anr-partition for which \(\bigcap\nolimits_i^r {\underline{\underline {}} } _1 \) convS _r is at least 1-dimensional.     

Pearson's correlation between three variables; using students’ basic knowledge of geometry for an exercise in mathematical statistics

The well known recurrence formulae for means and standard deviations have been extended to encompass bivariate data. The effectiveness of this technique is enhanced with the aid of a microcomputer.     

A generalization of Baer’s Lemma

There is a classical result known as Baer’s Lemma that states that an R-module E is injective if it is injective for R. This means that if a map from a submodule of R, that is, from a left ideal L of R to E can always be extended to R, then a map to E from a submodule A of any R-module B can be extended to B; in other words, E is injective. In this paper, we generalize this result to the category q _ω consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma is useful in proving that torsion free covers exist for q _ω.      

A Generalized Dade’s Lemma for Local Rings

We prove a generalized Dade’s Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient (and necessary) condition for the vanishing of all higher Ext or Tor of the modules. This condition involves the vanishing of all higher Ext or Tor of the modules over all quotients by a minimal generator of the ideal generated by the regular sequence.     

Robust Farkas’ lemma for uncertain linear systems with applications

We present a robust Farkas lemma, which provides a new generalization of the celebrated Farkas lemma for linear inequality systems to uncertain conical linear systems. We also characterize the robust Farkas lemma in terms of a generalized characteristic cone. As an application of the robust Farkas lemma we establish a characterization of uncertainty-immunized solutions of conical linear programming problems under uncertainty.     

Jacobson’s Lemma for Generalized Drazin–Riesz Inverses

Acta Mathematica Sinica, English Series - For bounded linear operators A, B, C and D on a Banach space X, we show that if BAC = BDB and CDB = CAC then I — AC is generalized Drazin—Riesz...     

Some new intuitionistic equivalents of Zorn’s Lemma

Two new intuitionistic equivalents to Zarns Lemma are stated and proved.     

Some generalizations of Hadamard’s-type inequalities through differentiability for s-convex functions and their applications

In this paper, a general form of integral inequalities of Hermite-Hadamard’s type through differentiability for s-convex function in second sense and whose all derivatives are absolutely continuous are established. The generalized integral inequalities contributes some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.     

Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

We consider the problem of minimizing a continuous function f over a compact set \({\mathbf {K}}\). We analyze a hierarchy of upper bounds proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), obtained by searching for an optimal probability density function h on \({\mathbf {K}}\) which is a sum of squares of polynomials, so that the expectation \(\int _{{\mathbf {K}}} f(x)h(x)dx\) is minimized. We show that the rate of convergence is no worse than \(O(1/\sqrt{r})\), where 2r is the degree bound on the density function. This analysis applies to the case when f is Lipschitz continuous and \({\mathbf {K}}\) is a full-dimensional compact set satisfying some boundary condition (which is satisfied, e.g., for convex bodies). The rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d, and if all moments of order up to \(2r+d\) of the Lebesgue measure on \({\mathbf {K}}\) are known, which holds, for example, if \({\mathbf {K}}\) is a simplex, hypercube, or a Euclidean ball.     

Students’ mathematical work on absolute value: focusing on conceptions,errors and obstacles

ZDM – Mathematics Education - This study investigates students’ conceptions of absolute value (AV), their performance in various items on AV, their errors in these items and the...     

Some finite generalizations of Euler’s pentagonal number theorem

Euler’s pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler’s pentagonal number theorem.