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Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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A DInduced Duality and Its Applications
, 2002
"... 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. Keywords: Convex cones, duality, bipolar theorem, conic optimization.
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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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.
ThB13.2 SelfConcordant Functions for Optimization on Smooth Manifolds
"... Abstract — 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 defin ..."
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Abstract — 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. I.
where f is a continuously differentiable function from ℜn +: = {x ∈ℜn: x ≥ 0} to ℜn and
, 2003
"... The homogeneous model for linear programs is an elegant means of obtaining the solution or certificate of infeasibility and has importance regardless of the method used for solving the problem, interiorpoint methods or other methods. In 1999, Andersen and Ye generalized this model to monotone compl ..."
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The homogeneous model for linear programs is an elegant means of obtaining the solution or certificate of infeasibility and has importance regardless of the method used for solving the problem, interiorpoint methods or other methods. In 1999, Andersen and Ye generalized this model to monotone complementarity problems (CPs) and showed that most of the desirable properties can be inherited as long as the problem is monotone. However, the strong dependence on monotonicity prevents the model from being extended to more general problems, such as P0 or P ∗ CPs. This paper presents a new homogeneous model and corresponding algorithm with the following features: (a) the homogeneous model preserves the P0 (P∗) property if the original problem is a P0 (P∗) CP, (b) the algorithm can be applied to P0 CPs starting at a positive point near the central trajectory and does not require any bigM penalty parameter, (c) the algorithm generates a sequence that approaches feasibility and optimality simultaneously for any P ∗ CP having a complementary solution, and (d) the algorithm solves P ∗ CPs with a strictly feasible point.