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22
LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 92 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
Making Prophecies with Decision Predicates
"... We describe a new algorithm for proving temporal properties expressed in LTL of infinitestate programs. Our approach takes advantage of the fact that LTL properties can often be proved more efficiently using techniques usually associated with the branchingtime logic CTL than they can with native LT ..."
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Cited by 7 (7 self)
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We describe a new algorithm for proving temporal properties expressed in LTL of infinitestate programs. Our approach takes advantage of the fact that LTL properties can often be proved more efficiently using techniques usually associated with the branchingtime logic CTL than they can with native LTL tools. The caveat is that, in certain instances, nondeterminism in the system’s transition relation can cause CTL methods to report counterexamples that are spurious with respect to the original LTL formula. To address this problem we describe an algorithm that, as it attempts to apply CTL proof methods, finds and then removes problematic nondeterminism via an analysis on the potentially spurious counterexamples. Problematic nondeterminism is characterized using decision predicates, and removed using a partial and symbolic determinization procedure that introduces new prophecy variables to predict the future outcome of these decisions. We demonstrate—using examples taken from the PostgreSQL database server, Apache web server, and Windows OS kernel—that our method can yield enormous performance improvements in comparison to known tools, allowing us to automatically prove properties of programs where we could not prove them before. 1.
Software Systems
"... Current C to gates synthesis tools do not support programs that make nontrivial use of dynamicallyallocated heap (e.g. linkedlist C programs that call malloc and free). The problem is that its difficult to determine an a priori bound on the amount of heap used during the program’s execution, if a ..."
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Cited by 6 (1 self)
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Current C to gates synthesis tools do not support programs that make nontrivial use of dynamicallyallocated heap (e.g. linkedlist C programs that call malloc and free). The problem is that its difficult to determine an a priori bound on the amount of heap used during the program’s execution, if a bound even exists. In this paper we develop a new method of synthesizing symbolic bounds expressed over generic parameters, thus leading to a C to gates synthesis flow for programs using dynamic allocation and deallocation. 1.
Abstractionrefinement for termination
 In 12th International Static Analysis Symposium(SAS’05
, 2005
"... In contrast to popular belief, proving ..."
Transposition theorems and qualificationfree optimality conditions
 SIAM J. Optimization
"... Abstract. New theorems of the alternative for polynomial constraints (based on the Positivstellensatz from real algebraic geometry) and for linear constraints (generalizing the transposition theorems of Motzkin and Tucker) are proved. Based on these, two KarushJohn optimality conditions – holding w ..."
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Cited by 4 (2 self)
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Abstract. New theorems of the alternative for polynomial constraints (based on the Positivstellensatz from real algebraic geometry) and for linear constraints (generalizing the transposition theorems of Motzkin and Tucker) are proved. Based on these, two KarushJohn optimality conditions – holding without any constraint qualification – are proved for single or multiobjective constrained optimization problems. The first condition applies to polynomial optimization problems only, and gives for the first time necessary and sufficient global optimality conditions for polynomial problems. The second condition applies to smooth local optimization problems and strengthens known local conditions. If some linear or concave constraints are present, the new version reduces the number of constraints for which a constraint qualification is needed to get the KuhnTucker conditions.
A Simple Algebraic Proof Of Farkas's Lemma And Related Theorems
, 1998
"... this paper we have given an alternative proof of Farkas's lemma, a proof that is based on a theorem, the main theorem, that relates to the eigenvectors of certain orthogonal matrices. This theorem is believed to be new, and the author is not aware of any similar theorem concerning orthogonal ma ..."
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Cited by 3 (0 self)
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this paper we have given an alternative proof of Farkas's lemma, a proof that is based on a theorem, the main theorem, that relates to the eigenvectors of certain orthogonal matrices. This theorem is believed to be new, and the author is not aware of any similar theorem concerning orthogonal matrices although he recently proved the weak form of the theorem using Tucker's theorem (see [5]). His proof of the theorem is "completely elementary" (a referee) and requires little more than a knowledge of matrix algebra for its understanding. Once the theorem is established, Tucker's theorem (via the Cayley transform), Farkas's lemma and many other theorems of the alternative follow trivially. Thus the paper establishes a connection between the eigenvectors of orthogonal matrices, duality in linear programming and theorems of the alternative that is not generally appreciated, and this may be of some theoretical interest.
Motzkin’s transposition theorem, and the related theorems of Farkas, Gordan and Stiemke
 Encyclopaedia of Mathematics, Supplement III
, 2002
"... Motzkin’s thesis [6], in particular his Transposition Theorem (Theorems 1–2 below), was a milestone in the development of linear inequalities and related areas. For two vectors u = (ui) and v = (vi) of equal dimension we denote by u ≧ v and u> v that the indicated inequality holds componentwise, ..."
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Cited by 3 (0 self)
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Motzkin’s thesis [6], in particular his Transposition Theorem (Theorems 1–2 below), was a milestone in the development of linear inequalities and related areas. For two vectors u = (ui) and v = (vi) of equal dimension we denote by u ≧ v and u> v that the indicated inequality holds componentwise, and by u � v the fact u ≧ v and u = v. Systems of linear inequalities appear in several forms; the following examples are typical: (a) A x ≦ b (b) A x = b, x ≧ 0 (c) A x ≦ b, Bx < c
A SimplexBased Extension of FourierMotzkin for Solving Linear Integer Arithmetic ⋆
"... Abstract. This paper describes a novel decision procedure for quantifierfree linear integer arithmetic. Standard techniques usually relax the initial problem to the rational domain and then proceed either by projection (e.g. OmegaTest) or by branching/cutting methods (branchandbound, branchandc ..."
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Cited by 1 (1 self)
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Abstract. This paper describes a novel decision procedure for quantifierfree linear integer arithmetic. Standard techniques usually relax the initial problem to the rational domain and then proceed either by projection (e.g. OmegaTest) or by branching/cutting methods (branchandbound, branchandcut, Gomory cuts). Our approach tries to bridge the gap between the two techniques: it interleaves an exhaustive search for a model with bounds inference. These bounds are computed provided an oracle capable of finding constant positive linear combinations of affine forms. We also show how to design an efficient oracle based on the Simplex procedure. Our algorithm is proved sound, complete, and terminating and is implemented in the altergo theorem prover. Experimental results are promising and show that our approach is competitive with stateoftheart SMT solvers. 1