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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 123 (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.
Preprocessing and Regularization for Degenerate Semidefinite Programs
, 2013
"... This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming r ..."
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Cited by 4 (0 self)
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This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primaldual interiorpoint, pd ip methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, wellposedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the BorweinWolkowicz facial reduction process to find a finite number, k, of rankrevealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, wellposed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The
Calculating the Cone of Directions of Constancy 1 H. WOLKOWICZ 2
"... Abstract. This note presents an algorithm that finds the cone of directions of constancy of a differentiable, faithfully convex function. Key Words. Cone of directions of constancy, faithfully convex functions, gradient. 1. ..."
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Abstract. This note presents an algorithm that finds the cone of directions of constancy of a differentiable, faithfully convex function. Key Words. Cone of directions of constancy, faithfully convex functions, gradient. 1.
NorthHolland Publishing Company CHARACTERIZATIONS OF OPTIMALITY WITHOUT CONSTRAINT QUALIFICATION FOR THE
, 1979
"... We consider the general abstract convex program (P) minimize [(x), subject to g(x) E S, where f is an extended convex functional on X, g: X ~ Y is Sconvex, S is a closed convex cone and X and Y are topological linear spaces. We present primal and dual characterizations for (P). These characterizat ..."
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We consider the general abstract convex program (P) minimize [(x), subject to g(x) E S, where f is an extended convex functional on X, g: X ~ Y is Sconvex, S is a closed convex cone and X and Y are topological linear spaces. We present primal and dual characterizations for (P). These characterizations are derived by reducing the problem to a standard Lagrange multiplier problem. Examples given include operator constrained problems as well as semiinfinite programming problems.
NorthHolland Publishing Company GEOMETRY OF OPTIMALITY CONDITIONS AND CONSTRAINT QUALIFICATIONS: THE CONVEX CASE*
, 1979
"... The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem. In addition, the "badly behaved set " of constraints, i.e. the set of constraints which ..."
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The cones of directions of constancy are used to derive: new as well as known optimality conditions; weakest constraint qualifications; and regularization techniques, for the convex programming problem. In addition, the "badly behaved set " of constraints, i.e. the set of constraints which causes problems in the KuhnTucker theory, is isolated and a computational procedure for checking whether a feasible point is regular or not is presented.
Method of Reduction in Convex Programming 1
"... Abstract. We present an algorithm which solves a convex program with faithfully convex (not necessarily differentiable) constraints. While finding a feasible starting point, the algorithm reduces the program to an equivalent program for which Slater's condition is satisfied. Included are algori ..."
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Abstract. We present an algorithm which solves a convex program with faithfully convex (not necessarily differentiable) constraints. While finding a feasible starting point, the algorithm reduces the program to an equivalent program for which Slater's condition is satisfied. Included are algorithms for calculating various objects which have recently appeared in the literature. Stability of the algorithm is discussed. Key Words. Convexity, subdifferentials, cones of directions of constancy, equality set of constraints, stability.