The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in 80s emphasis was on developing and implementing efficient variants of interior point methods for LP, the nineties have shown applicability to certain structured nonlinear programming and combinatorial problems. We will give a historical account of the developments and outline the major contributions to the field in the last decade. An important class of problems to which interior point methods are applicable is semidefinite optimization, which has recently gained much attention. It has a lot of applications in various fields (like control and system theory, combinatorial optimization, algebra, statistics, structural design) and can be efficiently solved with interior point methods.

This paper describes a parallel implementation of the primal-dual interior point method for a special class of large linear programs that occur in stochastic linear programming. The method used by Vanderbei and Carpenter [31] for removing dense columns is modified to eliminate variables which link blocks in stochastic linear programs. The algorithm developed was tested on six test problems from the Ho and Loute's collection of staircase linear programs [14] and on nine multi-scenario stochastic network problems [27] which arise in portfolio management. Numerical results demonstrate the efficiency of the resulting algorithm. For an 800 scenario problem with 91 constraints and 248 variables per scenario, a parallel efficiency of 99% is achieved. 1 Introduction

We present a primal-dual infeasible interior-point algorithm. As usual, the algorithm decreases the duality gap and the feasibility residuals at the same rate. Assuming that an optimal solution exists it is shown that at most O(n) iterations suffice to reduce the duality gap and the residuals by the factor 1/e. This implies an O(nlog(n/ε)) iteration bound for getting an ε-solution of the problem at hand, which coincides with the best known bound for infeasible interior-point algorithms. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. A special feature of the algorithm is that it uses only full-Newton steps. Two types of full-Newton steps are used, so-called feasibility steps and usual (centering) steps. Starting at strictly feasible iterates of a perturbed pair, (very) close its central path, feasibility steps serve to generate strictly feasible iterates for the next perturbed pair. By accomplishing a few centering steps for the new perturbed pair we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The algorithm finds an optimal solution or detects infeasibility or unboundedness of the given problem.

In this paper we use the interior point methodology to cover the main issues in linear programming: duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter. Key Words: linear programming, interior point methods, duality theory, parametric analysis, sensitivity analysis, primal-dual algorithm, implementation. Contents 1 Introduction 3 2 A new approach to the theory of linear programming 4 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Problem definition and assumptions : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 The logarithmic barrier approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3.1 The barrier problem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3.2 Minimizers of the barrier function : : : : : : : : : : : : : : : : : : : : : : : 6 2.3.3 The central path : : : : : : : : : ...

We study interior-point methods for optimization problems in the case of infeasibility or unboundedness. While many such methods are designed to search for optimal solutions even when they do not exist, we show that they can be viewed as implicitly searching for well-defined optimal solutions to related problems whose optimal solutions give certificates of infeasibility for the original problem or its dual. Our main development is in the context of linear programming, but we also discuss extensions to more general convex programming problems.

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Primal-dual Interior-Point Methods (IPMs) have shown their ability in solving large classes of optimization problems efficiently. Feasible IPMs require a strictly feasible starting point to generate the iterates that converge to an optimal solution. The self-dual embedding model provides an elegant solution to this problem with the cost of slightly increasing the size of the problem. On the other hand, Infeasible Interior Point Methods (IIPMs) can be initiated by any positive vector, and thus are popular in IPM softwares. In this paper we propose an adaptive large-update IIPM based on a specific self-regular proximity function, with barrier degree � 1 + log n, that operates in the infinity neighborhood of the central path. An O n 3 2 log n log n ɛ worst-case iteration bound of our new algorithm is established. This iteration bound improves the so far best O � n2 log n ɛ iterations bound of IIPMs

. Symmetricity of an optimal solution of SemiDefinite Program (SDP) with certain symmetricity is discussed based on symmetry property of the central path that is traced by a primaldual interior-point method. Introducing some operators for rearranging elements of matrices and vectors, three types of symmetric SDPs are defined by using those operators. The symmetricity of the solution on the central path is proved for each of symmetric SDPs. Therefore, it is theoretically guaranteed that a symmetric optimal solution is always obtained by using a primal-dual interiorpoint method even if there are other asymmetric optimal solutions. As an application of this result, we consider topology optimization problems of symmetric trusses that belong to one of the three types of symmetric SDPs, and we shall show that the symmetric optimal solution can be found regardless of the choice of member numbering and coordinate systems. Numerical experiments by using several algorithms for SDP illustrate rap...

The analytic center of a polytope P+ = fx x =1g is characterized by a saddle point condition m max g(x# y) = max g(x# y) # on the Lagrangian function log x j where A 2 R , and S++ = fx ? 0 x j =1g. This paper presents properties of the marginal function f(y)=maxfg(x# y):x 2 S++ g and explores the possibilities of a Lagrangian relaxation method for approximating the analytic center. 1.

Abstract. Motivated by a recent work of Setiono, a path-following algorithm for linear programming using both logarithmic and quadratic penalty functions is proposed. In the algorithm, a logarithmic and a quadratic penalty is placed on, respectively, the nonnegativity constraints and an arbitrary subset of the equality constraints; Newton’s method is applied to solve the penalized problem, and after each Newton step the penalty parameters are decreased. This algorithm maintains neither primal nor dual feasibility and does not require a Phase I. It is shown that if the initial iterate is chosen appropriately and the penalty parameters are decreased to zero in a particular way, then the algorithm is linearly convergent. Numerical results are also presented suggesting that the algorithm may be competitive with interior point algorithms in practice, requiring typically between 30-45 iterations to accurately solve each Netlib problem tested.