Interior-point Algorithm For Linear Programming Research Articles

Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

Interior-point algorithm for linear programming based on a new descent direction

We present a full-Newton step feasible interior-point algorithm for linear optimization based on a new search direction. We apply a vector-valued function generated by a univariate function on a new type of transformation on the centering equations of the system which characterizes the central path. For this, we consider a new function ψ(t) = t7\4. Furthermore, we show that the algorithm finds the ϵ-optimal solution of the underlying problem in polynomial time, namely O(√nlog(n+3\7√2)) iterations. Finally, a comparative numerical study is reported in order to analyze the efficiency of the proposed algorithm.

A step-truncated method in a wide neighborhood interior-point algorithm for linear programming

A step-truncated method in a wide neighborhood interior-point algorithm for linear programming

An Interior-Point Algorithm for Linear Programming with Optimal Selection of Centering Parameter and Step Size

For interior-point algorithms in linear programming, it is well-known that the selection of the centering parameter is crucial for proving polynomility in theory and for efficiency in practice. However, the selection of the centering parameter is usually by heuristics and separate from the selection of the line-search step size. The heuristics are quite different while developing practically efficient algorithms, such as MPC, and theoretically efficient algorithms, such as short-step path-following algorithm. This introduces a dilemma that some algorithms with the best-known polynomial bound are least efficient in practice, and some most efficient algorithms may not be polynomial. In this paper, we propose a systematic way to optimally select the centering parameter and line-search step size at the same time, and we show that the algorithm based on this strategy has the best-known polynomial bound and may be very efficient in computation for real problems.

Implementation of interior-point methods for LP based on Krylov subspace iterative solvers with inner-iteration preconditioning

We apply novel inner-iteration preconditioned Krylov subspace methods to the interior-point algorithm for linear programming (LP). Inner-iteration preconditioners recently proposed by Morikuni and Hayami enable us to overcome the severe ill-conditioning of linear equations solved in the final phase of interior-point iterations. The Krylov subspace methods do not suffer from rank-deficiency and therefore no preprocessing is necessary even if rows of the constraint matrix are not linearly independent. By means of these methods, a new interior-point recurrence is proposed in order to omit one matrix-vector product at each step. Extensive numerical experiments are conducted over diverse instances of 140 LP problems including the Netlib, QAPLIB, Mittelmann and Atomizer Basis Pursuit collections. The largest problem has 434,580 unknowns. It turns out that our implementation is more robust than the standard public domain solvers SeDuMi (Self-Dual Minimization), SDPT3 (Semidefinite Programming Toh-Todd-Tütüncü) and the LSMR iterative solver in PDCO (Primal-Dual Barrier Method for Convex Objectives) without increasing CPU time. The proposed interior-point method based on iterative solvers succeeds in solving a fairly large number of LP instances from benchmark libraries under the standard stopping criteria. The work also presents a fairly extensive benchmark test for several renowned solvers including direct and iterative solvers.

A Mehrotra type predictor-corrector interior-point algorithm for linear programming

In this paper, we analyze a feasible predictor-corrector linear programming variant of Mehrotra's algorithm. The analysis is done in the negative infinity neighborhood of the central path. We demonstrate the theoretical efficiency of this algorithm by showing its polynomial complexity. The complexity result establishes an improvement of factor $ n^3 $ in the theoretical complexity of an earlier presented variant in [2], which is a huge improvement. We examine the performance of our algorithm by comparing its implementation results to solve some NETLIB problems with the algorithm presented in [2].

Two computationally efficient polynomial-iteration infeasible interior-point algorithms for linear programming

Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bounds are not computationally efficient and algorithms demonstrated efficiency in computation do not have a good or any polynomial bound. Todd raised a question in 2002: “Can we find a theoretically and practically efficient way to reoptimize?” This paper is an effort to close the gap. We propose two arc-search infeasible interior-point algorithms with infeasible central path neighborhood wider than all existing infeasible interior-point algorithms that are proved to be convergent. We show that the first algorithm is polynomial and its simplified version has a complexity bound equal to the best known complexity bound for all (feasible or infeasible) interior-point algorithms. We demonstrate the computational efficiency of the proposed algorithms by testing all Netlib linear programming problems in standard form and comparing the numerical results to those obtained by Mehrotra’s predictor-corrector algorithm and a recently developed more efficient arc-search algorithm (the convergence of these two algorithms is unknown). We conclude that the newly proposed algorithms are not only polynomial but also computationally competitive compared to both Mehrotra’s predictor-corrector algorithm and the efficient arc-search algorithm.

A new infeasible-interior-point algorithm for linear programming over symmetric cones

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ1, τ2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r1.5loge−1) for the Nesterov-Todd (NT) direction, and O(r2loge−1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and e > 0 is the required precision. If starting with a feasible point (x0, y0, s0) in N(τ1, τ2, η), the complexity bound is $$O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)$$ for the NT direction, and O(rloge−1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.

A superlinearly convergent wide-neighborhood predictor–corrector interior-point algorithm for linear programming

In this paper we propose a new predictor-corrector algorithm with superlinear convergence in a wide neighborhood for linear programming problems. We let the centering parameter in a predictor step is chosen adaptively, which is different from other algorithms in the same wide neighborhood. The choice is a key for the local convergence of the new algorithm. In addition, we use the classical affine scaling direction as a part in a corrector step, not in a predictor step, which contributes to the complexity result. We prove that the new algorithm has a polynomial complexity of \(O(\sqrt{n}L)\), and the duality gap sequence is superlinearly convergent to zero, under the assumption that the iterate points sequence is convergent. Finally, numerical tests indicate its effectiveness.

A Homogeneous and Self-Dual Interior-Point Linear Programming Algorithm for Economic Model Predictive Control

We develop an efficient homogeneous and self-dual interior-point method (IPM) for the linear programs arising in economic model predictive control of constrained linear systems with linear objective functions. The algorithm is based on a Riccati iteration procedure, which is adapted to the linear system of equations solved in homogeneous and self-dual IPMs. Fast convergence is further achieved using a warm-start strategy. We implement the algorithm in MATLAB and C. Its performance is tested using a conceptual power management case study. Closed loop simulations show that: 1) the proposed algorithm is significantly faster than several state-of-the-art IPMs based on sparse linear algebra and 2) warm-start reduces the average number of iterations by 35%–40%.

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

Mehrotra's algorithm has been the most successful infeasible interior-point algorithm for linear programming since 1990. Most popular interior-point software packages for linear programming are based on Mehrotra's algorithm. This paper proposes an alternative algorithm, arc-search infeasible interior-point algorithm. We will demonstrate, by testing Netlib problems and comparing the test results obtained by arc-search infeasible interior-point algorithm and Mehrotra's algorithm, that the proposed arc-search infeasible interior-point algorithm is a more efficient algorithm than Mehrotra's algorithm.

A New Primal–Dual Predictor–Corrector Interior-Point Method for Linear Programming Based on a Wide Neighbourhood

In this paper, we propose a new predictor---corrector interior-point algorithm for linear programming based on a wide neighbourhood. In each iteration, the algorithm computes the Ai-Zhang's predictor direction (SIAM J. Optim. 16(2):400---417, 2005) and a new corrector direction, in an attempt to improve its performance. We drive that the duality gap reduces in both predictor and corrector steps. Moreover, we also prove that the complexity of the algorithm coincides with the best iteration bound for small neighbourhood algorithms. Finally, some numerical experiments are provided which reveal capability and effectiveness of the proposed method.

A Predictor-corrector algorithm with multiple corrections for convex quadratic programming

Recently an infeasible interior-point algorithm for linear programming (LP) was presented by Liu and Sun. By using similar predictor steps, we give a (feasible) predictor-corrector algorithm for convex quadratic programming (QP). We introduce a (scaled) proximity measure and a dynamical forcing factor (centering parameter). The latter is used to force the duality gap to decrease. The algorithm can decrease the duality gap monotonically. Polynomial complexity can be proved and the result coincides with the best one for LP, namely, $O(\sqrt{n}\log n\mu^{0}/\varepsilon)$ .

Exploiting special structure in semidefinite programming: A survey of theory and applications

Abstract Semidefinite programming (SDP) may be seen as a generalization of linear programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation. We survey three types of special structures in SDP data: 1. A common ‘chordal’ sparsity pattern of all the data matrices. This structure arises in applications in graph theory, and may also be used to deal with more general sparsity patterns in a heuristic way. 2. Low rank of all the data matrices. This structure is common in SDP relaxations of combinatorial optimization problems, and SDP approximations of polynomial optimization problems. 3. The situation where the data matrices are invariant under the action of a permutation group, or, more generally, where the data matrices belong to a low dimensional matrix algebra. Such problems arise in truss topology optimization, particle physics, coding theory, computational geometry, and graph theory. We will give an overview of existing techniques to exploit these structures in the data. Most of the paper will be devoted to the third situation, since it has received the least attention in the literature so far.

A Full-Newton Step Infeasible Interior-Point Algorithm for Linear Programming Based on a Kernel Function

This paper proposes an infeasible interior-point algorithm with full-Newton step for linear programming, which is an extension of the work of Roos (SIAM J. Optim. 16(4):1110–1136, 2006). The main iteration of the algorithm consists of a feasibility step and several centrality steps. We introduce a kernel function in the algorithm to induce the feasibility step. For parameter p∈[0,1], the polynomial complexity can be proved and the result coincides with the best result for infeasible interior-point methods, that is, O(nlog n/e).

Enlarging neighborhoods of interior-point algorithms for linear programming via least values of proximity measure functions

It is well known that a wide-neighborhood interior-point algorithm for linear programming performs much better in implementation than its small-neighborhood counterparts. In this paper, we provide a unified way to enlarge the neighborhoods of predictor–corrector interior-point algorithms for linear programming. We prove that our methods not only enlarge the neighborhoods but also retain the so-far best known iteration complexity and superlinear (or quadratic) convergence of the original interior-point algorithms. The idea of our methods is to use the global minimizers of proximity measure functions.

A New Notion of Weighted Centers for Semidefinite Programming

The notion of weighted centers is essential in V-space interior-point algorithms for linear programming. Although there were some successes in generalizing this notion to semidefinite programming via weighted center equations, we still do not have a generalization that preserves two important properties---(1) each choice of weights uniquely determines a pair of primal-dual weighted centers, and (2) the set of all primal-dual weighted centers completely fills up the relative interior of the primal-dual feasible region. This paper presents a new notion of weighted centers for semidefinite programming that possesses both uniqueness and completeness. Furthermore, it is shown that under strict complementarity, these weighted centers converge to weighted centers of optimal faces. Finally, this convergence result is applied to homogeneous cone programming, where the central paths defined by a certain class of optimal barriers for homogeneous cones are shown to converge to analytic centers of optimal faces in the presence of strictly complementary solutions.

Stabilization of Mehrotra's primal–dual algorithm and its implementation

In this paper we apply a stabilization procedure proposed by Kovačević-Vujčić and Ašić to the Mehrotra's primal–dual interior-point algorithm for linear programming. Transformations of the dual problem corresponding to the stabilization procedure are considered. The stabilization procedure and Mehrotra's algorithm are implemented in the package MATHEMATICA. A number of highly degenerate test examples are used to compare the modified Mehrotra's method with respect to the original one and the codes PCx, LIPSOL and MOSEK. We also provide numerical results on some examples from the Netlib test set.

A New Iteration-Complexity Bound for the MTY Predictor-Corrector Algorithm

In this paper we present a new iteration-complexity bound for the Mizuno--Todd--Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min{cTx:Ax = b, x \ge 0} with decision variable $x\in \Re^n$, e show that the MTY P-C algorithm, started from a well-centered interior-feasible solution with duality gap $n \mu_0$, finds an interior-feasible solution with duality gap less than $n\eta$ in ${\cal O}(T(\mu_0/\eta)+n^{3.5}\log({{\bar\chi^*_A}}))$ iterations, where $T(t)\equiv\min\{n^2\log(\log t) ,\log t\}$ for all $t > 0$ and ${{\bar\chi^*_A}}$ is a scaling invariant condition number associated with the matrix A. More specifically, ${{\bar\chi^*_A}}$ is the infimum of all the conditions numbers ${\bar\chi}_{AD}$, where D varies over the set of positive diagonal matrices. Under the setting of the Turing machine model, our analysis yields an ${\cal O}(n^{3.5}L_A+ \min\{n^2\log L,L\})$ iteration-complexity bound for the MTY P-C algorithm to find a primal-dual optimal solution, where LA and L are the input sizes of the matrix A and the data (A,b,c), respectively. This contrasts well with the classical iteration-complexity bound for the MTY P-C algorithm, which depends linearly on L instead of log L.