Infeasible Interior-point Algorithm Research Articles

Superlinear convergence of an interior point algorithm on linear semi-definite feasibility problems

In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction, to solve semi-definite programs (SDPs) is shown by (i) assuming that the given SDP is nondegenerate and making modifications to these algorithms [Kojima et al. Local convergence of predictor–corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Math. Program. Ser. A 80 (1998), pp. 129–160], or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems (LSDFPs) and requiring the initial iterate to the algorithm to satisfy certain conditions [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621]. Otherwise, these algorithms are not easy to implement even though they are shown to have polynomial iteration complexities and superlinear convergence [Luo et al. Superlinear convergence of a symmetric primal–dual path following algorithm for semidefinite programming, SIAM J. Optim. 8 (1998), pp. 59–81]. The conditions in [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621] that the initial iterate to the algorithm is required to satisfy to have superlinear convergence when solving LSDFPs however are not practical. In this paper, we propose a practical initial iterate to an implementable infeasible interior point algorithm that guarantees superlinear convergence when the algorithm is used to solve the homogeneous feasibility model of an LSDFP.

A full-Newton step infeasible interior point algorithm and its parameters analysis

A full-Newton step infeasible interior point algorithm and its parameters analysis

A full-Newton step infeasible interior-point algorithm based on a kernel function with a new barrier term

A full-Newton step infeasible interior-point algorithm based on a kernel function with a new barrier term

A limiting analysis on regularization of singular SDP and its implication to infeasible interior-point algorithms

We consider primal-dual pairs of semidefinite programs and assume that they are singular, i.e., both primal and dual are either weakly feasible or weakly infeasible. Under such circumstances, strong duality may break down and the primal and dual might have a nonzero duality gap. Nevertheless, there are arbitrary small perturbations to the problem data which would make them strongly feasible thus zeroing the duality gap. In this paper, we conduct an asymptotic analysis of the optimal value as the perturbation for regularization is driven to zero. Specifically, we fix two positive definite matrices, I_p and I_d, say, (typically the identity matrices), and regularize the primal and dual problems by shifting their associated affine space by eta I_p and varepsilon I_d, respectively, to recover interior feasibility of both problems, where varepsilon and eta are positive numbers. Then we analyze the behavior of the optimal value of the regularized problem when the perturbation is reduced to zero keeping the ratio between eta and varepsilon constant. A key feature of our analysis is that no further assumptions such as compactness or constraint qualifications are ever made. It will be shown that the optimal value of the perturbed problem converges to a value between the primal and dual optimal values of the original problems. Furthermore, the limiting optimal value changes “monotonically” from the primal optimal value to the dual optimal value as a function of theta , if we parametrize (varepsilon , eta ) as (varepsilon , eta )=t(cos theta , sin theta ) and let trightarrow 0. Finally, the analysis leads us to the relatively surprising consequence that some representative infeasible interior-point algorithms for SDP generate sequences converging to a number between the primal and dual optimal values, even in the presence of a nonzero duality gap. Though this result is more of theoretical interest at this point, it might be of some value in the development of infeasible interior-point algorithms that can handle singular problems.

A polynomial time infeasible interior-point arc-search algorithm for convex optimization

This paper proposes an infeasible interior-point algorithm for the convex optimization problem using arc-search techniques. The proposed algorithm simultaneously selects the centering parameter and the step size, aiming at optimizing the performance in every iteration. Analytic formulas for the arc-search are provided to make the arc-search method very efficient. The convergence of the algorithm is proved and a polynomial bound of the algorithm is established. The preliminary numerical test results indicate that the algorithm is efficient and effective

A New Projection Contraction Algorithm for Semidefinite Programming

This paper studies a projective contraction algorithm for solving semidefinite programming problems. The interior point algorithm is a classic algorithm for solving semidefinite programming problems. It can be divided into feasible and infeasible algorithms according to whether the initial point is feasible or not. The feasible interior point algorithm needs to use a self-dual strategy to find strictly feasible points. Although the infeasible interior point algorithm does not require the initial point to be strictly feasible, the theoretical analysis of the algorithm is very complicated. Compared with the interior point algorithm, the projection contraction algorithm has no requirement on the initial point, and the algorithm analysis is much simpler. The optimal condition of the semidefinite programming problem can be rewritten as a projection equation, which is then solved by the projection contraction algorithm. We propose a new projection contraction algorithm based on different descent directions to solve the semidefinite programming problem. Further, under the framework of the projection contraction algorithm, an improved algorithm is proposed, that is, changing the descending direction by solving the linear equation system, thus avoiding the high-dimensional inversion operation so that the algorithm can solve the large-scale semidefinite programming problem. Numerical experiments show that the algorithm is feasible and the improved algorithm is competitive with the interior point algorithm.

An Infeasible Interior-point Algorithm for Monotone Linear Complementarity Problems

In this study, we implement a variant of infeasible interior-point algorithm for solving monotone linear complementarity problems (LCP). We first reformulate the monotone LCP as an minimization problem. Then a descent iterative method is applied to the latter. The descent direction is computed via the Newton method. However, for maintaining the positivity of iterates, a novel and efficient strategy is proposed. Some numerical results are reported to show the efficiency of our proposed approach.

A Polynomial-Iteration Infeasible Interior-Point Algorithm with Arc-Search for Semidefinite optimization

A Polynomial-Iteration Infeasible Interior-Point Algorithm with Arc-Search for Semidefinite optimization

An infeasible interior-point arc-search algorithm for nonlinear constrained optimization

In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search since they compute a next iterate on a straight line determined by a search direction which approximates the central path. The proposed arc-search interior-point algorithm uses an arc for the approximation. We discuss convergence properties of the proposed algorithm. We also conduct numerical experiments on the CUTEst benchmark problems and compare the performance of the proposed arc-search algorithm with that of a line-search algorithm. Numerical results indicate that the proposed arc-search algorithm reaches the optimal solution using fewer iterations but longer times than a line-search algorithm. A modification that leads to a faster arc-search algorithm is also discussed.

A Second-Order Corrector Infeasible Interior-Point Method for Semidefinite Optimization Based on a Wide Neighborhood

This paper proposes a second-order corrector infeasible interior-point method for semidefinite optimization in a large neighborhood of the central path. Our algorithm uses the Nesterov–Todd search directions as a Newton direction. We make use of the scaled Newton directions for symmetric search directions. Based on Ai and Zhang idea, we decompose the Newton directions into two orthogonal directions corresponding to positive and negative parts of the right-hand side of the Newton equation. In such a way, we use different step lengths for each of them and the corrector step is multiplied by the square of the step length of the infeasible directions in the expression of the new iterate. Starting with a point $$(X^0, y^0, S^0)$$ in the neighborhood in terms of Frobenius norm, we show that the algorithm terminates after at most $$\mathcal {O}(n^{\frac{5}{4}}\log \varepsilon ^{-1})$$ iterations, improving by a factor $$n^{\frac{1}{4}}$$ results on these kinds of algorithms. We believe that this is the first infeasible interior-point algorithm in a large neighborhood, which uses a Newton direction decomposition and a second-order corrector step to improve the iteration complexity. The main difference of the proposed method comparing with the existing methods in literature is the strategy for generating corrector directions. Some preliminary numerical results are given to verify the efficiency of the algorithm.

A New Predictor-corrector Infeasible Interior-point Algorithm for Linear Optimization in aWide Neighborhood

In this paper, we propose a Mizuno-Todd-Ye type predictor-corrector infeasible interior-point method for linear optimization based on a wide neighborhood of the central path. According to Ai-Zhang’s original idea, we use two directions of distinct and orthogonal corresponding to the negative and positive parts of the right side vector of the centering equation of the central path. In the predictor stage, the step size along the corresponded infeasible directions to the negative part is chosen. In the corrector stage by modifying the positive directions system a full-Newton step is removed. We show that, in addition to the predictor step, our method reduces the duality gap in the corrector step and this can be a prominent feature of our method. We prove that the iteration complexity of the new algorithm is 𝒪(n log ɛ−1), which coincides with the best known complexity result for infeasible interior-point methods, where ɛ > 0 is the required precision. Due to the positive direction new system, we improve the theoretical complexity bound for this kind of infeasible interior-point method [1] by a factor of n . Numerical results are also provided to demonstrate the performance of the proposed algorithm.

An arc-search infeasible interior-point method for semidefinite optimization with the negative infinity neighborhood

We present an arc-search infeasible interior-point algorithm for semidefinite optimization using the Nesterov-Todd search directions. The algorithm is based on the negative infinity neighborhood of the central path. The algorithm searches an e-approximate solution of the problem along the ellipsoidal approximation of the entire central path. The convergence analysis of the algorithm is presented and shows that the algorithm has the iteration complexity bound $\mathcal {O}\big (n^{3/2}\log {\varepsilon }^{-1}\big )$ . Here, n is the dimension of the problem and e is the required precision. The numerical results show that our algorithm is efficient and promising.

Iterative Descent Method for Generalized Leontief Model

In this article, we show that under certain condition the generalized Leontief model is solvable by iterative descent method based on infeasible interior point algorithm. We prove the convergence of the method from strictly positive starting point. A numerical example is presented to demonstrate the performance of the algorithm.

A wide neighborhood predictor-infeasible corrector interior-point algorithm for linear optimization

In this paper, we propose a theoretical framework of a predictor-corrector interior-point method for linear optimization based on the one-norm wide neighborhood of the central path, focusing on infeasible corrector steps. Here, we call the predictor-infeasible corrector algorithm. At each iteration, the proposed algorithm computes an infeasible corrector step in addition to the Ai-Zhang search directions and considers the Newton direction as a linear combination of these directions. We represent the complexity analysis of the algorithm and conclude that its iteration bound is $${\mathcal {O}}(n\log \varepsilon ^{-1})$$ . To our knowledge, this is the best complexity result up to now for infeasible interior-point methods based on these kinds of search directions. The complexity bound obtained here is the same as small neighborhood infeasible interior point algorithms.

A Full-NT Step Infeasible Interior-Point Algorithm for Mixed Symmetric Cone LCPs

An infeasible interior-point algorithm for mixed symmetric cone linear complementarity problems is proposed. Using the machinery of Euclidean Jordan algebras and Nesterov-Todd search direction, the convergence analysis of the algorithm is shown and proved. Moreover, we obtain a polynomial time complexity bound which matches the currently best known iteration bound for infeasible interior-point methods.

A regularized interior-point method for constrained linear least squares

We propose an infeasible interior-point algorithm for constrained linear least-squares problems based on the primal-dual regularization of convex programs of Friedlander and Orban. Regularization allows us to dispense with the assumption that the active gradients are linearly independent. At each iteration, a linear system with a symmetric quasi-definite (SQD) matrix is solved. This matrix is shown to be uniformly bounded and nonsingular. While the linear system may be solved using sparse LDL factorization, we observe that other approaches may be used. In particular, we build on the connection between SQD linear systems and unconstrained linear least-squares problems to solve the linear system with sparse QR factorization. We establish conditions under which a solution of the original, constrained least-squares problem is recovered. We report computational experience with the sparse QR factorization and illustrate the potential advantages of our approach.

Infeasible interior-point algorithms based on sampling average approximations for a class of stochastic complementarity problems and their applications

This paper considers a class of stochastic complementarity problems (SCP). Different from the classical complementarity problems, the SCP contains a mathematical expectation, which may not be evaluated in an explicit form in general. We combine an interior-point algorithm for deterministic cases with the well-known sample average approximation (SAA) techniques to present an SAA-based infeasible interior-point algorithm for the SCP. We investigate the convergence properties and computational complexity of the proposed algorithm under mild assumptions. Then, we extend these results to a class of mixed SCPs. Furthermore, we apply the proposed algorithms to solve a stochastic natural gas transmission problem and a stochastic oligopoly model. Preliminary numerical experiments indicate that the proposed approach is competitive with some existing methods. • We study a class of stochastic complementarity problems. • We present an infeasible interior-point algorithm for the considered problem. • We investigate convergence properties and computational complexity of the algorithm. • We apply the proposed approach to natural gas transmission and oligopoly problems.

An Infeasible Interior-Point Algorithm for Stochastic Second-Order Cone Optimization

Alzalg (J Optim Theory Appl 163(1):148–164, 2014) derived a homogeneous self-dual algorithm for stochastic second-order cone programs with finite event space. In this paper, we derive an infeasible interior-point algorithm for the same stochastic optimization problem by utilizing the work of Rangarajan (SIAM J Optim 16(4), 1211–1229, 2006) for deterministic symmetric cone programs. We show that the infeasible interior-point algorithm developed in this paper has complexity less than that of the homogeneous self-dual algorithm mentioned above. We implement the proposed algorithm to show that they are efficient.

A modified infeasible interior-point algorithm with full-Newton step for semidefinite optimization

ABSTRACTRecently, Mansouri et al. (J. Optim. Theory Appl. 166: 605-618, 2015) presented an improved infeasible interior-point algorithm for linear optimization. Their algorithm has the shortcoming that the proximity measure may be still large when the duality gap approaches to zero. In this paper, we propose an infeasible interior-point algorithm for semidefinite optimization with a modified search direction. This modification is an attempt to decrease the value of the proximity measure, which is important to determine whether or not to perform centreing steps in the classical infeasible interior-point algorithms. Some preliminary numerical results show the benefit of the proposed algorithm as well.

Simplified infeasible interior-point algorithm for linear optimization based on a simple function

Based on a similar kernel function, we present an infeasible version of the interior-point algorithm for linear optimization introduced by Wang et al. (2016). The property of exponential convexity is still important to simplify the analysis of the algorithm. The iteration bound coincides with the currently best iteration bound for infeasible interior-point algorithms.