The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.

Abstract. The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and the Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when a semidefinite program to be solved is large scale and sparse.

In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.

. A primal-dual infeasible-interior-point path-following algorithm is proposed for solving semidefinite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithms finds an optimal solution in at most O( p nL) iterations, where n is the size of the problem and L is the logarithm of the ratio of the initial error and the tolerance. If the starting point is large enough then the algorithm terminates in at most O(nL) steps either by finding a solution or by determining that the primal-dual problem has no solution of norm less than a given number. Moreover, we propose a sufficient condition for the superlinear convergence of the algorithm. In addition, we give two special cases of SDP for which the algorithm is quadratically convergent. Key words. semidefinite programming, path-following, infeasible-interior-point algorithm, polynomiality, superlinear convergence. AMS ...

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 infeasible start predictor-corrector algorithm for semidefinite programming is proposed. It is a direct extension of the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming. The algorithm uses the Kojima-Shindoh-Hara/HelmbergRendl -Vanderbei-Wolkowicz/Monteiro direction in the predictor step and the Alizadeh-Haeberly-Overton direction in the corrector step. It has polynomial complexity for general problems and is superlinearly convergent with Q-order at least 1.5 under strict complementarity and nondegeneracy conditions.

A simple homogeneous primal-dual feasibility model is proposed for semidefinite programming (SDP) problems. Two infeasible-interior-point algorithms are applied to the homogeneous formulation. The algorithms do not need big M initialization. If the original SDP problem has a solution, then both algorithms find an ffl-approximate solution (i.e., a solution with residual error less than or equal to ffl) in at most O( p n ln(ae ffl 0 =ffl)) steps, where ae is the trace norm of a solution and ffl 0 is the residual error at the (normalized) starting point. A simple way of monitoring possible infeasibility of the original SDP problem is provided such that in at most O( p n ln(aeffl 0 =ffl)) steps either an ffl-approximate solution is obtained, or it is determined that there is no solution with trace norm less than or equal to a given number ae ? 0. Key Words: semidefinite programming, homogeneous interior-point algorithm, polynomial complexity. Abbreviated Title: Homogeneous al...

this paper. 1.9.1 The spectral bundle method

In this paper we study interior point trajectories in semidefinite programming (SDP) including the central path of an SDP. This work was inspired by the seminal work by Megiddo on linear programming trajectories [15]. Under an assumption of primal and dual strict feasibility, we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of, respectively, the primal and the dual problems. We consider a class of trajectories that are similar to the central path, but can be constructed to pass through any given interior feasible point and study their convergence. Finally, we study the first order derivatives of these trajectories and their convergence. We also consider higher order derivatives associated with these trajectories.

In this paper we present a generalization of the predictor corrector method of linear programming problem to semidefinite linear programming problem. We consider a direction which, we show, belongs to a family of directions presented by Kojima, Shindoh and Hara, and, one of the directions analyzed by Monteiro. We show that starting with the initial complementary slackness violation of t 0 , in O(jlog( ffl t 0 )j p n) iterations of the predictor corrector method, the complementary slackness violation can be reduced to less than or equal to ffl ? 0. We also analyze a modified corrector direction in which the linear system to be solved differs from that of the predictor in only the right hand side, and obtain a similar bound. We then use this modified corrector step in an implementable method which is shown to take a total of O(jlog( ffl t 0 )j p nlog(n)) predictor and corrector steps. Key words: Linear programming, Semidefinite programming, Interior point methods, Path following, ...