SeDuMi is an add-on for MATLAB, that lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.

informs doi 10.1287/moor.1060.0242

How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the two-phase approach or using the so-called big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the self-dual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded self-dual problem. The embedded self-dual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the so-called stri...

In this paper we study the properties of the analytic central path of a semide nite programming problem under perturbation of a set of input parameters. Speci cally, we analyze the behavior of solutions on the central path with respect to changes on the right hand side of the constraints, including the limiting behavior when the central optimal solution is approached. Our results are of interest for the sake ofnumerical analysis, sensitivity analysis and parametric programming. Under the primal-dual Slater condition and the strict complementarity condition we show that the derivatives of central solutions with respect to the right hand side parameters converge as the path tends to the central optimal solution. Moreover, the derivatives are bounded, i.e. a Lipschitz constant exists. This Lipschitz constant can be thought of as a condition number for the semide nite programming problem. It is a generalization of the familiar condition number for linear equation systems and linear programming problems. However, the generalized condition number depends on the right hand side parameters as well, whereas it is well-known that in the linear programming case the condition number depends only on the constraint matrix. We demonstrate that the existence of strictly complementary solutions is important for the Lipschitz constant to exist. Moreover, we give an example in which the set of right hand side parameters for which the strict complementarity condition holds is neither open nor closed. This is remarkable since a similar set for which the primal-dual Slater condition holds is always open. Key words: analytic central path, semide nite programming, sensitivity, condition number.

In this paper we introduce a conic optimization formulation for inequality-constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. Keywords: Convex Programming, Convex Cones, Self-Dual Embedding, Self-Concordant Barrier Functions. # Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported by Hong Kong RGC Earmarked Grants CUHK4181/00E and CUHK4233/01E. 1 1

For iterative sequences that converge to the solution set of a linear matrix inequality, we show that the distance of the iterates to the solution set is at most O(ffl 2 \Gammad ). The nonnegative integer d is the so--called degree of singularity of the linear matrix inequality, and ffl denotes the amount of constraint violation in the iterate. For infeasible linear matrix inequalities, we show that the minimal norm of ffl--approximate primal solutions is at least 1=O(ffl 1=(2 d \Gamma1) ), and the minimal norm of ffl--approximate Farkas-- type dual solutions is at most O(1=ffl 2 d \Gamma1 ). As an application of these error bounds, we show that for any bounded sequence of ffl--approximate solutions to a semi-definite programming problem, the distance to the optimal solution set is at most O(ffl 2 \Gammak ), where k is the degree of singularity of the optimal solution set. Keywords: semi-definite programming, error bounds, linear matrix inequality, regularized duality. AMS s...

In this paper Lipschitzian type error bounds are derived for general convex conic problems under various regularity conditions. Specifically, it is shown that if the recession directions satisfy Slater's condition then a global Lipschitzian type error bound holds. Alternatively, if the feasible region is bounded, then the ordinary Slater condition guarantees a global Lipschitzian type error bound. These can be considered as generalizations of previously known results for inequality systems. Moreover, some of the results are also generalized to the intersection of multiple cones. Under Slater's condition alone, a global Lipschitzian type error bound may not hold. However, it is shown that such an error bound holds for a specific region. For linear systems we show that the constant involved in Hoffman's error bound can be estimated by the so-called condition number for linear programming. Key words: Error bound, convex conic problems, LMIs, condition number. AMS subject classification: 5...

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