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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.
Note on a paper of Broyden
, 1999
"... Recently Broyden [1] proved a property of orthogonal matrices from which he derived Farkas' lemma and some related results. It is shown that Broyden's result straightforwardly follows from wellknown theorems of the alternative, like Motzkin's transposition theorem and Tucker's t ..."
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Recently Broyden [1] proved a property of orthogonal matrices from which he derived Farkas' lemma and some related results. It is shown that Broyden's result straightforwardly follows from wellknown theorems of the alternative, like Motzkin's transposition theorem and Tucker's theorem, which are all logically equivalent to Farkas' lemma; we also answer the question of Broyden on how to efficiently compute the sign matrix of an orthogonal matrix. Key words: Orthogonal matrices, theorems of the alternative, transposition theorem, analytic center 1 Introduction Farkas' Lemma [4] is one of the theorems of the alternative; these theorems have a long history and it is wellknown that they are all equivalent in the sense that these theorems can easily be derived from each other. See, e.g., [6]. Using an appealing metaphor of Broyden [1] one may say that they resemble cities situated on a high plateau; travel between them is not too difficult, the hard part is the initial ascent from the pl...
Fakult"at f"ur Mathematik, Universit"at Wien
"... February 20, 2006 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 cond ..."
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February 20, 2006 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.
1 CONSTRUCTIVE PROOF OF FARKAS ’ LEMMA IMPERVIOUS TO DEGENERACY
, 2003
"... and Hanson does indeed constitute by itself a new constructive proof of Farkas’ lemma (in primal and dual form), we apply it to solve the restricted feasibility subproblems that occur in two primaldual activeset linear programming algorithms, thus establishing the relationship with a recently pro ..."
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and Hanson does indeed constitute by itself a new constructive proof of Farkas’ lemma (in primal and dual form), we apply it to solve the restricted feasibility subproblems that occur in two primaldual activeset linear programming algorithms, thus establishing the relationship with a recently proposed (dated in October, 2002) primaldual activeset scheme. As a result, the proposed algorithms are impervious to degeneracy, both in Phase I and in Phase II, and finite in exact arithmetic without having to resort to any anticycling pivoting rule, and hence they are by themselves constructive proofs of the strong duality theorem. Finally, we describe the sparse implementation of both algorithms in terms of certain sparse Cholesky factorization.