In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasible-primal-dual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasible-primal-dual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...

A modification of the (infeasible) primal-dual interior point method is developed. The method uses multiple corrections to improve the centrality of the current iterate. The maximum number of corrections the algorithm is encouraged to make depends on the ratio of the efforts to solve and to factorize the KKT systems. For any LP problem, this ratio is determined right after preprocessing the KKT system and prior to the optimization process. The harder the factorization, the more advantageous the higher-order corrections might prove to be. The computational performance of the method is studied on more difficult Netlib problems as well as on tougher and larger real--life LP models arising from applications. The use of multiple centrality corrections gives on the average a 25% to 40% reduction in the number of iterations compared with the widely used second-order predictor-corrector method. This translates into 20% to 30% savings in CPU time.

We present a generalization of a homogeneous self-dual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "big-M" parameter or two-phase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interior-point and infeasible-starting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and self-dual, infeasible-starting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK-5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...

In this paper we study a first-order and a high-order algorithm for solving linear complementarity problems. These algorithms are implicitly associated with a large neighborhood whose size may depend on the dimension of the problems. The complexity of these algorithms depends on the size of the neighborhood. For the first order algorithm, we achieve the complexity bound which the typical large-step algorithms possess. It is well-known that the complexity of large-step algorithms is greater than that of short-step ones. By using high-order power series (hence the name high-order algorithm), the iteration complexity can be reduced. We show that the complexity upper bound for our high-order algorithms is equal to that for short-step algorithms. Key Words: Interior point algorithm, High-order power series, Large neighborhood, Large step, Complexity, Linear complementarity problem. Abbreviated Title: Interior point algorithms based on large neighborhoods AMS(MOS) subject classifications: 90...

Abstract. A higher order corrector-predictor interior-point method is proposed for solving sufficient linear complementarity problems. The algorithm produces a sequence of iterates in the N − ∞ neighborhood of the central path. The algorithm does not depend on the handicap κ of the problem. It has O((1 + κ) √ nL) iteration complexity and is superlinearly convergent even for degenerate problems. Key words. neighborhood linear complementarity, interior-point, path-following, corrector-predictor, wide AMS subject classifications. 90C51, 90C33 1. Introduction. The MTY predictor-corrector algorithm proposed by Mizuno, Todd and Ye [9] is a typical representative of a large class of MTY type predictorcorrector methods, which play a very important role among primal-dual interior point methods. It was the first algorithm for linear programming (LP) that had both polynomial complexity and superlinear convergence. This result was extended to monotone

In this paper, we first introduce the notion of self-regular functions. Various appealing properties of self-regular functions are explored and we also discuss the relation between selfregular functions and the well-known self-concordant functions. Then we use such functions to define self-regular proximity measure for path-following interior point methods for solving linear optimization (LO) problems. Any self-regular proximity measure naturally defines a primal-dual search direction. In this way a new class of primal-dual search directions for solving LO problems is obtained. Using the appealing properties of self-regular functions, we prove that these new large-update path-following methods for LO enjoy a polynomial, O n q+1 2q log n iteration bound, where q ≥ 1 is the so-called barrier degree of the self-regular ε proximity measure underlying the algorithm. When q increases, this � bound approaches the √n n best known complexity bound for interior point methods, namely O log. Our unified �√n ε n analysis provides also the O log best known iteration bound of small-update IPMs. ε At each iteration, we need only to solve one linear system. As a byproduct of our results, we remove some limitations of the algorithms presented in [24] and improve their complexity as well. An extension of these results to semidefinite optimization (SDO) is also discussed.

We propose a new class of primal-dual methods for linear optimization (LO). By using some new analysis tools, we prove that the large update method for LO based on the new search direction has a polynomial complexity O i n 4 4+ae log n " j iterations where ae 2 [0; 2] is a parameter used in the system defining the search direction. If ae = 0, our results reproduce the well known complexity of the standard primal dual Newton method for LO. At each iteration, our algorithm needs only to solve a linear equation system. An extension of the algorithms to semidefinite optimization is also presented. Keywords: Linear Optimization, Semidefinite Optimization, Interior Point Method, PrimalDual Newton Method, Polynomial Complexity. AMS Subject Classification: 90C05 1 Introduction Interior point methods (IPMs) are among the most effective methods for solving wide classes of optimization problems. Since the seminal work of Karmarkar [7], many researchers have proposed and analyzed various ...

In this paper we discuss the polynomiality of a feasible version of Mehrotra’s predictor-corrector algorithm whose variants have been widely used in several IPM based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms in this algorithm may lead to small steps being taken in order to keep the iterates in a large neighborhood of the central path, which is important to proving polynomial complexity properties of this method. Motivated by this example, we introduce a safeguard in Mehrtora’s algorithm that keeps the iterates in the prescribed neighborhood and allows us to obtain a positive lower bound on the step size. This safeguard strategy is also used when the affine scaling direction performs poorly. We prove that the safeguarded algorithm will terminate after at most O(n2 log (x0) T s0 ɛ) iteration. By modestly modifying the corrector direction, we reduce the iteration complexity to O(n log (x0) T s0 ɛ). To en-sure fast asymptotic convergence of the algorithm, we changed Mehrotra’s updating scheme of the centering parameter slightly while keeping the safeguard. The new al-gorithms have the same order of iteration complexity as the safeguarded algorithms, but enjoy superlinear convergence as well. Numerical results using the McIPM and LIPSOL software packages are reported.

Recently, in [10], the authors presented a new large-update primal-dual method for Linear Optimization, whose O(n 2 3 log n " ) iteration bound substantially improved the classical bound for such methods, which is O n log n " . In this paper we present an improved analysis of the new method. The analysis uses some new mathematical tools, partially developed in [11], where we consider a whole family of interior-point methods which contains the method considered in this paper. The new analysis yields an O p n log n log n " iteration bound for large-update methods. Since we concentrate on one specic member of the family considered in [11], the analysis is signicantly simpler than in [11]. The new bound further improves the iteration bound for large-update methods, and is quite close to the currently best iteration bound known for interior-point methods, namely O p n log n " . Hence, the existing gap between the iteration bounds for small-update and large-update met...

In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen, Roos and Terlaky enhances an asymptotical O( p nL) complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semi-definite linear complementarity problems.