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11
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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SelfConcordant Functions for Optimization on Smooth Manifolds
, 2004
"... This paper discusses selfconcordant functions on smooth manifolds. In Euclidean space, this class of functions are utilized extensively in interiorpoint methods for optimization because of the associated low computational complexity. Here, the selfconcordant function is carefully defined on a di ..."
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Cited by 9 (3 self)
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This paper discusses selfconcordant functions on smooth manifolds. In Euclidean space, this class of functions are utilized extensively in interiorpoint methods for optimization because of the associated low computational complexity. Here, the selfconcordant function is carefully defined on a differential manifold. First, generalizations of the properties of selfconcordant functions in Euclidean space are derived. Then, Newton decrement is defined and analyzed on the manifold that we consider. Based on this, a damped Newton algorithm is proposed for optimization of selfconcordant functions, which guarantees that the solution falls in any given small neighborhood of the optimal solution, with its existence and uniqueness also proved in this paper, in a finite number of steps. It also ensures quadratic convergence within a neighborhood of the minimal point. This neighborhood can be specified by the the norm of Newton decrement. The computational complexity bound of the proposed approach is also given explicitly. This complexity bound is O( − ln(ɛ)), where ɛ is the desired precision. An interesting optimization problem is given to illustrate the proposed concept and algorithm.
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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Cited by 3 (2 self)
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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.
Using interiorpoint methods within an outer approximation framework for mixedinteger nonlinear programming
 IMAMINLP Issue
"... Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via o ..."
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Cited by 2 (0 self)
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Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via outer approximation. However, traditionally, infeasible primaldual interiorpoint methods have had two main perceived deficiencies: (1) lack of infeasibility detection capabilities, and (2) poor performance after a warmstart. In this paper, we propose the exact primaldual penalty approach as a means to overcome these deficiencies. The generality of this approach to handle any change to the problem makes it suitable for the outer approximation framework, where each nonlinear subproblem can differ from the others in the sequence in a variety of ways. Additionally, we examine cases where the nonlinear subproblems take on special forms, namely those of secondorder cone programming problems and semidefinite programming problems. Encouraging numerical results are provided.
A Dinduced duality and its applications
"... This paper attempts to extend the notion of duality for convex cones, by basing it on a predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the innerproduct is replaced by a prespecified con ..."
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This paper attempts to extend the notion of duality for convex cones, by basing it on a predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the innerproduct is replaced by a prespecified conic ordering, defined by a convex cone D, and the innerproduct itself is replaced by a general multidimensional bilinear mapping. This new type of duality is termed the Dinduced duality in the paper. Basic properties of the extended duality, including the extended bipolar theorem, are proven. Examples are given to show the applications of the new results.
ON IMPLEMENTATION OF A SELFDUAL EMBEDDING METHOD FOR CONVEX PROGRAMMING∗
, 2004
"... In this paper, we implement Zhang’s method [22], which transforms a general convex optimization problem with smooth convex constraints into a convex conic optimization problem and then apply the techniques of selfdual embedding and central path following for solving the resulting conic optimizatio ..."
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In this paper, we implement Zhang’s method [22], which transforms a general convex optimization problem with smooth convex constraints into a convex conic optimization problem and then apply the techniques of selfdual embedding and central path following for solving the resulting conic optimization model. A crucial advantage of the approach is that no initial solution is required, and the method is particularly suitable when the feasibility status of the problem is unknown. In our implementation, we use a merit function approach proposed by Andersen and Ye [1] to determine the step size along the search direction. We evaluate the efficiency of the proposed algorithm by observing its performance on some test problems, which include logarithmic functions, exponential functions and quadratic functions in the constraints. Furthermore, we consider in particular the geometric programming and Lpprogramming problems. Numerical results of our algorithm on these classes of optimization problems are reported. We conclude that the algorithm is stable, efficient and easytouse in general. As the method allows the user to freely select the initial solution if he/she so wishes, it is natural to take advantage of this and apply the socalled warmstart strategy, whenever the data of a new problem is not too much different from a previously solved problem. This strategy turns out to be effective, according to our numerical experience.
AND THE COMMITTEE ON GRADUATE STUDIES
, 2004
"... dissertation for the degree of Doctor of Philosophy. ..."
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Wire Sizing, Repeater Insertion and Dominant Time Constant Optimization for a Bus Line.
, 2004
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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 ..."
Abstract
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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...
prohibited without the written consent of the copyright owner.
, 2004
"... Wire sizing, repeater insertion and dominant time constant optimization for a bus line ..."
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Wire sizing, repeater insertion and dominant time constant optimization for a bus line