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

It is known that predictor-corrector methods in a large neighborhood of the central path are among the most efficient interior-point methods (IPMs) for linear optimization (LO) problems. The best iteration bound based on the classical logarithmic barrier function is O(nlog (n/∊)). In this paper, we propose a family of self-regular proximity-based predictor-corrector (SRPC) IPMs for LO in a large neighborhood of the central path. In the predictor step, we use either an affine scaling or a self-regular direction; in the corrector step, we use always a self-regular direction. Our new algorithms use a special proximity function with different search directions and thus allows us to improve the so far best theoretical iteration complexity for a family of SRPC IPMs. An O\((\sqrt{n}{\exp} ((1 - q + {\log} n)/2) {\log} n {\log} (n/\epsilon))\) worst-case iteration bound with quadratic convergence is established, where q is the barrier degree of the SR proximity function.

An interior-point algorithm for linear optimization based on a new barrier function

Interior-point algorithm for linear optimization based on a new trigonometric kernel function

We present several improvements of the full-Newton step infeasible interior-point method for linear optimization introduced by Roos (SIAM J. Optim. 16(4):1110---1136, 2006). Each main step of the method consists of a feasibility step and several centering steps. We use a more natural feasibility step, which targets the μ+-center of the next pair of perturbed problems. As for the centering steps, we apply a sharper quadratic convergence result, which leads to a slightly wider neighborhood for the feasibility steps. Moreover, the analysis is much simplified and the iteration bound is slightly better.

The Karmarkar interior point algorithm has made it possible to solve large-scale decision problems that previously could not be solved in reasonable time, or were too large to be solved at all. In this paper, the authors present the results obtained from using the KORBX Advanced Mathematical Programming System (KMPS), which uses Karmarkar's interior point algorithm, to solve a number of linear optimization problems arising from a long-term fuel planning problem. Comparison with the results obtained using software based on the simplex method demonstrates the drastic improvements in solution time for the interior point method, especially with increase in problem sizes. This confirms earlier comparisons of interior point and simplex methods. The paper includes preliminary ideas and results on ways to combine the interior point and simplex methods in order to benefit from the superior speed performance of the former, and the warm-start and hot-start capabilities of the latter methods. >

This paper presents a new interior point quadratic programming algorithm which can solve power system optimization problems with significantly less computational efforts. The proposed algorithm has the following two special features. First, it is based on the path-following interior point algorithm whose search direction is the Newton direction, and therefore the algorithm has quadratic convergence. In the second place, it solves directly a symmetric indefinite system and thus the algorithm avoids the formation of [AD/sup -1/A/sup T/] and as a result generates fewer fill-ins than the case of factorizing the positive definite system matrix for large scale systems. This has brought about a profound speed-up. Since the formula of the interior point method have been deduced more generally, the proposed algorithm can start from either a feasible (interior point) or an infeasible point (noninterior point). Numerical results on the IEEE test systems and a Japanese 344 bus system have verified that the proposed algorithm possesses enough robustness and needs significantly less solution time compared with already reported applications of the interior point method.

We present an analysis of the full-Newton step infeasible interior-point algorithm for semidefinite optimization, which is an extension of the algorithm introduced by Roos [C. Roos, A full-Newton step 𝒪(n) infeasible interior-point algorithm for linear optimization, SIAM J. Optim. 16 (2006), pp. 1110–1136] for the linear optimization case. We use the proximity measure σ(V) = ‖I − V 2‖ to overcome the difficulty of obtaining an upper bound of updated proximity after one full-Newton step, where I is an identity matrix and V is a symmetric positive definite matrix. It turns out that the complexity analysis of the algorithm is simplified and the iteration bound obtained is improved slightly.

In this paper, we propose a new long-step interior point method for solving sufficient linear complementarity problems. The new algorithm combines two important approaches from the literature: the main ideas of the long-step interior point algorithm introduced by Ai and Zhang and the algebraic equivalent transformation technique proposed by Darvay. Similar to the method of Ai and Zhang, our algorithm also works in a wide neighborhood of the central path and has the best known iteration complexity of short-step variants. However, due to the properties of the applied transforming function in Darvay’s technique, the wide neighborhood definition in the analysis depends on the value of the handicap. We implemented not only the theoretical algorithm but a greedy variant of the new method (working in a neighborhood independent of the handicap) in MATLAB and tested its efficiency on both sufficient and non-sufficient problem instances. In addition to presenting our numerical results, we also make some interesting observations regarding the analysis of Ai–Zhang type methods.

Kernel functions serve the central goal of creating new search directions for the primal-dual interiorpoint algorithm to solve linear optimization problems. A significantly improved primal-dual interior-point algorithm for linear optimization is presented based on a novel kernel function. We show a primal-dual interior-point technique for linear optimization based on a class of kernel functions that are eligible. This research presents a new efficient kernel function-based primal-dual IPM algorithm for semidefinite programming problems based on the Nesterov-Todd (NT) direction. With a new and simple technique, we propose a new kernel function to obtain an optimal solution of the perturbed problem (SDP)µ. We obtain the best-known complexity results,for smalland large-update namely O(pp+1/2p √nlog tr(X0S0)/ε) and O((pn)p+1/2p log trX0S0/ε) large update To prove the effectiveness of our proposed kernel function, we compare our numerical results with some alternatives presented by Touil et al. (2017).

A full-Newton step [formula omitted] infeasible-interior-point algorithm for linear complementarity problems

This paper proposes a nonlinear optimization interior point (IP) method for the determination of maximum loadability in a power system. Details of the implementation of pure primal-dual and predictor-corrector primal-dual IP algorithms are presented. It is shown that most of the computational effort of the algorithm is taken by the formation and factorization of the augmented Hessian matrix of the IP algorithm. The size of this matrix can be as large as ten times the number of buses in the system. Comparisons of the two IP implementations with large scale power systems with as many as 4000 buses are presented. It is shown that the IP algorithm constitutes an effective method for the determination of the maximum loadability in a power system.

Based on an equivalent reformulation of the central path, we obtaina modified-Newton step for linear optimization. Using this step, wepropose an infeasible interior-point algorithm. The algorithm usesonly one full-modified-Newton step search in each iteration. Thecomplexity bound of the algorithm is the best known for infeasibleinterior-point algorithm.

An interior-point algorithm for linear optimization based on a new algebraically equivalent transformation

In this paper we propose a new class of primal-dual path-following interior point algorithms for solving monotone linear complementarity problems. At each iteration, the method would select a target on the central path with a large update from the current iterate, and then the Newton method is used to get the search directions, followed by adaptively choosing the step sizes, which are, e.g., the largest possible steps before leaving a neighborhood that is as wide as the given ${\cal N}^-_{\infty}$ neighborhood. The only deviation from the classical approach is that we treat the classical Newton direction as the sum of two other directions, corresponding to, respectively, the negative part and the positive part of the right-hand side. We show that if these two directions are equipped with different and appropriate step sizes, then the method enjoys the low iteration bound of $O(\sqrt{n}\log L)$, where n is the dimension of the problem and $L=\frac{(x^0)^Ts^0}{\ep}$ with $\ep$ the required precision and $(x^0,s^0)$ the initial interior solution. For a predictor-corrector variant of the method, we further prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of $1-1/O(\sqrt{n})$. Additionally, if the problem has a strict complementary solution, then the predictor steps converge Q-quadratically.

As recently demonstrated by the study of the primal-dual interior-point methods based on kernel functions, a kernel function not only serves to determine the search direction and measure the distance of the current iteration point to the µ-center, but also affects the iteration complexity and the practical computational efficiency of the algorithm.In this paper, we propose a primal-dual interior-point method for a linear optimization based on a new parameterized kernel function.The construction of the new parameterized kernel function is motivated by the parameterized ways of existing kernel functions.By using properties of the new parameterized kernel function, we improve the iteration bound of the large-update method from O(n, which is the best theoretical iteration result currently known.Finally, some numerical results are given to present the efficiency and potential of our kernel function.