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A Unifying Investigation of InteriorPoint Methods for Convex Programming
 FACULTY OF MATHEMATICS AND INFORMATICS, TU DELFT, NL2628 BL
, 1992
"... In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interiorp ..."
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Cited by 5 (4 self)
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In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interiorpoint methods for these programs can thus be reduced to the analysis of interiorpoint methods with selfconcordant barrier functions.
On Conically Ordered Convex Programs
, 2003
"... In this paper we study a special class of convex optimization problems called conically ordered convex programs (COCP), where the feasible region is given as the level set of a vectorvalued nonlinear mapping, expressed as a nonnegative combination of convex functions. The nonnegativity of the vect ..."
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In this paper we study a special class of convex optimization problems called conically ordered convex programs (COCP), where the feasible region is given as the level set of a vectorvalued nonlinear mapping, expressed as a nonnegative combination of convex functions. The nonnegativity of the vectors is defined using a predescribed conic ordering. The new model extends the ordinary convex programming models where the feasible sets are the level sets of convex functions, and it also extends the famous linear conic optimization models. We introduce a condition on the barrier function for the orderdefining cone, termed as the coneconsistent property. The relationship between the coneconsistent barriers and the selfconcordance barriers is investigated. We prove that if the orderdefining cone admits a selfconcordant and coneconsistent barrier function, and moreover, if the constraint functions are all convex quadratic then the overall composite barrier function is selfconcordant. The problem is thus solvable in polynomial time, following Nesterov and Nemirovskii, by means of the pathfollowing method. If the constraint functions are not quadratic, but harmonically convex, then we propose a variant of IriImai type potential reduction method. In addition to the selfconcordance and the coneconsistence conditions, we assume that the barrier function for the orderdefining cone has the property that the image of the cone under its Hessian matrix is contained in its dual cone. All these conditions are satisfied by the familiar selfscaled cones. Under these conditions we show that the IriImai type potential reduction algorithm converges in polynomial time. Duality issues related to this class of optimization problems are discussed as well.
The Netherlands.
"... In this paper we introduce a potential reduction method for harmonically convex programming. We show that, if the objective function and the m constraint functions are all kharmonically convex in the feasible set, then the number of iterations needed to find an optimal solution is bounded by a pol ..."
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In this paper we introduce a potential reduction method for harmonically convex programming. We show that, if the objective function and the m constraint functions are all kharmonically convex in the feasible set, then the number of iterations needed to find an optimal solution is bounded by a polynomial in m, k and log(1/). The method requires either the optimal objective value of the problem or an upper bound of the harmonic constant k as a working parameter. Moreover, we discuss the relation between the harmonic convexity condition used in this paper and some other convexity and smoothness conditions used in the literature.
Let
, 2003
"... In this paper we study a special class of convex optimization problems called conically ordered convex programs (COCP), where the feasible region is given as the level set of a vectorvalued nonlinear mapping, expressed as a nonnegative combination of convex functions. The nonnegativity of the vecto ..."
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In this paper we study a special class of convex optimization problems called conically ordered convex programs (COCP), where the feasible region is given as the level set of a vectorvalued nonlinear mapping, expressed as a nonnegative combination of convex functions. The nonnegativity of the vectors is defined using a predescribed conic ordering. The new model extends the ordinary convex programming models where the feasible sets are the level sets of convex functions, and it also extends the famous linear conic optimization models. We introduce a condition on the barrier function for the orderdefining cone, termed as the coneconsistent property. The relationship between the coneconsistent barriers and the selfconcordance barriers is investigated. We prove that if the orderdefining cone admits a selfconcordant and coneconsistent barrier function, and moreover, if the constraint functions are all convex quadratic then the overall composite barrier function is selfconcordant. The problem is thus solvable in polynomial time, following Nesterov and Nemirovski, by means of the pathfollowing method. If the constraint functions are not quadratic, but harmonically convex, then we propose a variant of IriImai type potential reduction method. To facilitate the analysis, in addition to the selfconcordance and the coneconsistence conditions, we assume that the barrier function for the orderdefining cone is so that the image of the cone under its Hessian matrix is contained in the dual cone. All these conditions are satisfied by the familiar selfscaled cones. Under these conditions we show that the IriImai type potential reduction algorithm converges in polynomial time. Duality issues related to this class of optimization problems are discussed as well. Keywords: co...
A Value Estimation Approach to the IriImai Method for Constrained Convex Optimization
, 2003
"... In this paper, we propose an extension of the socalled IriImai method to solve constrained convex programming problems. The original IriImai method is designed for linear programs and assumes that the optimal objective value of the optimization problem is known in advance. Zhang [18] extends the ..."
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In this paper, we propose an extension of the socalled IriImai method to solve constrained convex programming problems. The original IriImai method is designed for linear programs and assumes that the optimal objective value of the optimization problem is known in advance. Zhang [18] extends the method for constrained convex optimization, but the optimum value is still assumed to be known in advance. In our new extension this last requirement on the optimal value is relaxed; instead, only a lower bound of the optimal value is needed. Our approach uses a multiplicative barrier function for the problem with a univariate parameter that represents an estimated optimum value of the original optimization problem. An optimal solution to the original problem can be traced down by minimizing the multiplicative barrier function. Due to the convexity of this barrier function, the optimal objective value as well as the optimal solution of the original problem, are sought iteratively by applying Newton's method to the multiplicative barrier function. A new formulation of multiplicative barrier function is further developed to acquire computational tractability and e#ciency. Numerical results are presented to show the e#ciency of the new method. Keywords: Constrained Convex Optimization, IriImai's Algorithm, Value Estimation. # Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong.