Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization

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).

Based on a new parametric kernel function, this paper presents a primal-dual large-update interior-point algorithm (IPM) for semi-definite optimization (SDO) problems. The new parametric function is neither self-regular function nor the usual logarithmic barrier function. It is strongly convex and possesses some novel analytic properties. We analyse this new parametric kernel function and show that the proposed algorithm has favorable complexity bound in terms of the analytic properties of the kernel function. Moreover, the complexity bound for our large-update IPM is shown to be O(\sqrt{n}(\log n)^2 \log\frac{n}{\epsilon}). Some numerical results are reported to illustrate the feasibility of the proposed algorithm.

In this paper we propose a class of new large-update primal-dual interior-point algorithms for P *(κ) nonlinear complementarity problem (NCP), which are based on a class of kernel functions investigated by Bai et al. in their recent work for linear optimization (LO). The arguments for the algorithms are followed as Peng et al.’s for P *(κ) complementarity problem based on the self-regular functions [Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, Princeton, 2002]. It is worth mentioning that since this class of kernel functions includes a class of non-self-regular functions as special case, so our algorithms are different from Peng et al.’s and the corresponding analysis is simpler than theirs. The ultimate goal of the paper is to show that the algorithms based on these functions have favorable polynomial complexity.

A class of polynomial primal-dual interior-point algorithms for P * (κ ) linear complementarity problems (LCPs) are presented. We generalize Ghami et al.'s[A polynomial-time algorithm for linear optimization based on a new class of kernel functions(2008)] algorithm for linear optimization (LO) problem to P * (κ ) LCPs. Our analysis is based on a class of finite kernel functions which have the linear and quadratic growth terms. Since P *(κ ) LCP is a generalization of LO problem, we lose the orthogonality of the vectors dx and ds . So our analysis is different from the one in Ghami et al's algorithm. Despite this, the favorable complexity result is obtained, namely, $O((1 + 2\kappa)n^{\frac{1}{1+p}}\log n \log(n/\epsilon))$, which is better than the usual large-update primal-dual algorithm based on the classical logarithmic barrier function for P * (κ ) LCP.

Exploring complexity of large update interior-point methods for [formula omitted] linear complementarity problem based on Kernel function

In the present paper we present a class of polynomial primal-dual interior-point algorithms for semidefinite optimization based on a kernel function. This kernel function is not a so-called self-regular function due to its growth term increasing linearly. Some new analysis tools were developed which can be used to deal with complexity analysis of the algorithms which use analogous strategy in [5] to design the search directions for the Newton system. The complexity bounds for the algorithms with large- and small-update methods were obtained, namely, O(qn(p+q)/q(p+1)) log n/ɛ and O(q2 √n) log n/ɛ, respectively.

This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidef-inite systems known to be strongly factorizable to a form of Cholesky-like factorization.Two different regularization techniques,primal; and dual, are very well suited to the (infeasible) primal-dual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs.

In this paper, we present a primal-dual interior point method for linear optimization problems based on a new Kernel function with on a new parameterized logarithmic barrier term. We prove that the proposed kernel function belongs to the eligible class. We derive the complexity bounds for large and small-update methods respectively.

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm for solving linear optimization problems. In this paper we present a new kernel function which yields an algorithm with the best known complexity bound for both large- and small-update methods.

A primal-dual interior point algorithm for solving general nonlinear programming problems is presented. The algorithm solves the perturbed optimality conditions by applying a quasi-Newton method, where the Hessian of the Lagrangian is replaced by a positive definite approximation. An approximation of Fletcher's exact and differentiable merit function together with line-search procedures are incorporated into the algorithm. The line-search procedures are used to modify the length of the step so that the value of the merit function is always reduced. Different step-sizes are used for the primal and dual variables. The search directions are ensured to be descent for the merit function, which is thus used to guide the algorithm to an optimum solution of the constrained optimisation problem. The monotonic decrease of the merit function at each iteration, ensures the global convergence of the algorithm. Finally, preliminary numerical results demonstrate the efficient performance of the algorithm for a variety of problems.

In this paper a primal-dual path-following interior-point algorithm for the monotone semidefinite linear complementarity problem is presented. The algorithm is based on Nesterov-Todd search directions and on a suitable proximity for tracing approximately the central-path. We provide an unified analysis for both long and small-update primal-dual algorithms. Finally, the iteration bounds for these algorithms are obtained.

The traditional load-shedding models usually use global optimization to get the load-shedding region, which will cause multiple variables, huge computing scale and other problems. This makes it hard to meet the requirements of timeliness in on-line power system operation risk assessment. In order to solve the problems of the present load-shedding models, a load-shedding model based on sensitivity analysis is proposed in this manuscript. By calculating the sensitivity of each branch on each bus, the collection of buses which have remarkable influence on reducing the power flow on over-load branches is obtained. In this way, global optimization is turned to local optimization, which can narrow the solution range. By comprehensively considering the importance of load bus and adjacency principle regarding the electrical coupling relationship, a load-shedding model is established to get the minimum value of the load reduction from different kinds of load buses, which is solved by the primal dual interior point algorithm. In the end, different cases on the IEEE 24-bus, IEEE 300-bus and other multi-node systems are simulated. The correctness and effectiveness of the proposed load-shedding model are demonstrated by the simulation results.

This paper presents a new primal-dual interior-point algorithm with reduced potential function for a class of convex programming, based on the ideas of that method for solving linear programming. The new algorithm chooses the classical Newton direction as iteration direction and its iteration stepsize is determined by potential function. As the search directions Δxand Δsaren't orthogonal any more, the complexity analysis of this method is different from that of linear programming, correspondingly. Under a scaled Lipschitz condition, the algorithm is proved to possess O(nL) iteration-complexity bounds

The introduction of kernel function in primal-dual interior point methods represents not only a measure of the distance between the iterate and the central path, but also plays an important role in the amelioration of the computational complexity of an interior point algorithm. In this work, we present a primal-dual interior point method for linear optimization problems based on a new kernel function with an efficient logarithmic barrier term. We derive the complexity bounds for large and small-update methods respectively. We obtain the best known complexity bound for large-update given by Peng et al., which improves significantly the so far obtained complexity results based on a logarithmic kernel function given by El Ghami et al. The results obtained in this paper are the first with respect to the kernel function with a logarithmic barrier term.

: A globally convergent primal-dual predictor corrector interior point algorithm for linear programming has been developed.