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An InteriorPoint Method For Convex Fractional Programming
 AT&T Bell Labs Numerical Analysis Manuscript
, 1993
"... We present an interiorpoint method for convex fractional programming. The proposed algorithm converges in polynomial time, just as in the case of a convex problem, even though convex fractional programs are only pseudoconvex. More precisely, the rate of convergence is measured in terms of the area ..."
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We present an interiorpoint method for convex fractional programming. The proposed algorithm converges in polynomial time, just as in the case of a convex problem, even though convex fractional programs are only pseudoconvex. More precisely, the rate of convergence is measured in terms of the area of twodimensional convex sets C k containing the optimal points, and the area of C k is reduced by a constant factor c ! 1 at each iteration. The factor c depends only on the selfconcordance parameter of a barrier function associated with the feasible set. We present an outline of a practical implementation of the proposed method, and we report results of a few numerical experiments. 1. Introduction Interiorpoint methods for the solution of nonlinear programming problems were already introduced in the 1950s and 1960s; see [6] and the references given there. In the 1970s, new and seemingly superior approaches, such as sequential quadratic programming techniques, were developed, and as a ...
A PredictorCorrector Algorithm For A Class Of Nonlinear Saddle Point Problems
 SIAM Journal on Control and Optimization
, 1994
"... . An interior pathfollowing algorithm is proposed for solving the nonlinear saddle point problem minimax c T x + OE(x) + b T y \Gamma /(y) \Gamma y T Ax subject to (x; y) 2 X \Theta Y ae R n \Theta R m ; where OE(x) and /(y) are smooth convex functions and X and Y are boxes (hyperrecta ..."
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. An interior pathfollowing algorithm is proposed for solving the nonlinear saddle point problem minimax c T x + OE(x) + b T y \Gamma /(y) \Gamma y T Ax subject to (x; y) 2 X \Theta Y ae R n \Theta R m ; where OE(x) and /(y) are smooth convex functions and X and Y are boxes (hyperrectangles). This problem is closely related to models in stochastic programming and optimal control studied by Rockafellar and Wets. Existence conditions on a central path are established. Starting from an initial solution near the central path with duality gap O(¯), the algorithm finds an ffloptimal solution of the problem in O( p m+ nj log ¯=fflj) iterations if both OE(x) and /(y) satisfy a scaled Lipschitz condition. Keywords. Interior point methods, optimal control, saddle point problem, stochastic programming. Abbreviated title. IP method for saddle point problems AMS subject classifications. 49J35, 65K10, 90C06, 90C15, 90C33 October, 1994 This research is partially supported by grant...
A Short Survey on Ten Years Interior Point Methods
, 1995
"... The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability to c ..."
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The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability to certain structured nonlinear programming and combinatorial problems. We will give a historical account of the developments and outline the major contributions to the field in the last decade. An important class of problems to which interior point methods are applicable is semidefinite optimization, which has recently gained much attention. It has a lot of applications in various fields (like control and system theory, combinatorial optimization, algebra, statistics, structural design) and can be efficiently solved with interior point methods.
A new linesearch step based on the Weierstrass function for minimizing a class of logarithmic barrier functions
, 1994
"... This article is concerned with linesearch procedures for a class of problems with certain nonlinear constraints. The class includes as special cases linear and convex quadratic programming problems, entropy programming problems and minimization problems over the cone of positive semidefinite matric ..."
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This article is concerned with linesearch procedures for a class of problems with certain nonlinear constraints. The class includes as special cases linear and convex quadratic programming problems, entropy programming problems and minimization problems over the cone of positive semidefinite matrices, [1, 2, 18, 7]. For solving a constrained optimization problem of the form minff 0 (x) j f i (x) 0 for 1 i mg (1.1) with four times continuously differentiable functions f i , we consider logarithmic barrier functions of the form