In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see [9]). The class of problems under consideration includes linear programming, semidefinite programming and quadratically constrained quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.

This note summarizes a report with the same title, where a study was carried out regarding a unified approach, proposed by Kojima, Mizuno and Yoshise, for interior point algorithms for the linear complementarily problem with a positive semi-definite matrix. This approach is extended to nonsymmetric matrices with nonnegative principal minors.

This paper deals with a class of primal-dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara [11]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X 1=2 S \Gamma1=2 . They study three primaldual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno and Yoshise [10]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present: 1) a simplified polynomial convergence proof for one of their short-step path-following ...

In this paper a symmetric primal-dual transformation for positive semidefinite programming is proposed. For standard SDP problems, after this symmetric transformation the primal variables and the dual slacks become identical. In the context of linear programming, existence of such a primal-dual transformation is a well known fact. Based on this symmetric primal-dual transformation we derive Newton search directions for primal-dual path-following algorithms for semidefinite programming. In particular, we generalize: (1) the short step path following algorithm, (2) the predictor-corrector algorithm and (3) the largest step algorithm to semidefinite programming. It is shown that these algorithms require at most O( p n j log ffl j) main iterations for computing an ffl-optimal solution. The symmetric primal-dual transformation discussed in this paper can be interpreted as a specialization of the scaling-point concept introduced by Nesterov and Todd [12] for self-scaled conic problems. The ...

A smooth approximation p(x; ff) to the plus function: maxfx; 0g, is obtained by integrating the sigmoid function 1=(1 + e \Gammaffx ), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for ff sufficiently large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite ff. Speedup over MINOS 5.4 was as high as 515 times for linear inequalities of size 1000 \Theta 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems are converted into a system of smooth nonlinear equations and are solved by a quadratically convergent Newton method. For monotone LCP's with as many as 400 variables, the proposed approach was as much as 85 times faster than Lemke's method. Key Words: Smo...

This paper establishes the polynomialconvergence of the class of primal-dual feasible interiorpoint algorithms for semidefinite programming (SDP) based on Monteiro and Zhang family of search directions. In contrast to Monteiro and Zhang's work, no condition is imposed on the scaling matrix that determines the search direction. We show that the polynomial iterationcomplexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SDP. Since Monteiro and Zhang family of directions includes the Alizadeh, Haeberly and Overton direction, we establish for the first time the polynomial convergence of algorithms based on this search direction. Keywords: Semidefinite programming, interior-point methods, polynomial complexity, pathfollowing methods, primal-dual methods. AMS 1991 subject classification: 65K05, 90C25, 90C...

. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno-Todd-Ye type predictorcorrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno-Todd-Ye type predictor-corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, Infeasible-Interior-Point Method, Predictor-CorrectorMethod, Superlinear Convergence, Primal-Dual Nondegeneracy Abbreviated Title. Interior-Point Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japa...

A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y- ffx)). Under the assumption that the mapping f is a P,,-function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x, y) = t(a, b) and (x, y) 8 0 until the parameter t> 0 attains 0. Here (a, b) denotes a 2n-dimensional constant positive vector. We establish the existence of a trajectory which leads to a solution of the problem, and then present a numerical method for tracing the trajectory. We also discuss the global and local convergence of the method.

This paper proposes a globally convergent predictor-corrector infeasible-interior-point algorithm for the monotone semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction, and shows its quadratic local convergence under the strict complementarity condition.

The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence properties for algorithms based on Newton iterations. This problem provides a simple and general framework for most existing primal-dual interior point methods. The conclusion is that most of the published algorithms of this kind generate convergent sequences. In many cases (whenever the convergence is not too fast in a certain sense), the sequences converge to the analytic center of the optimal face.