Interior-point Method For Semidefinite Optimization Research Articles

A new wide-neighborhood predictor-corrector interior-point method for semidefinite optimization

A new wide-neighborhood predictor-corrector interior-point method for semidefinite optimization

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.

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.

A new primal-dual interior-point method for semidefinite optimization based on a parameterized kernel function

As indicated in the recent studies about primal-dual interior-point methods (IPMs) 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 $$\mu $$ -center, but also affects the iteration complexity and the practical computational efficiency of the algorithm. In this paper, we propose a new IPM for semidefinite optimization (SDO) based on a parameterized kernel function which is a generalization of the one presented by Bai et al. (Optim Methods Softw 17(6):985–1008, 2002). By using the good properties of the parameterized kernel function, we deduce that the iteration bound for large-update method is $$O(\sqrt{n}\log {n}\log {\frac{n}{\epsilon }})$$ for $$q=O(n)$$ , which is the best known complexity results for such methods. In our knowledge, this result is the first instance of primal-dual interior point method for SDO which involving the kernel function. Some numerical results have been provided.

Complexity analysis of infeasible interior-point method for semidefinite optimization based on a new trigonometric kernel function

In this paper, a full Nesterov–Todd step infeasible interior-point method for solving semidefinite optimization problems based on a new kernel function is analyzed. In each iteration, the algorithm involves a feasibility step and several centrality steps. The centrality step is focused on Nesterov–Todd search directions, while we used a kernel function with trigonometric barrier term to induce the feasibility step. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

New complexity analysis of a full Nesterov–Todd step interior-point method for semidefinite optimization

In this paper, we propose a new primal-dual path-following interior-point method for semidefinite optimization based on a new reformulation of the nonlinear equation of the system which defines the central path. The proposed algorithm takes only full Nesterov and Todd steps and therefore no line-searches are needed for generating the new iterations. The convergence of the algorithm is established and the complexity result coincides with the best-known iteration bound for semidefinite optimization problems.

A primal–dual interior-point method for semidefinite optimization based on a class of trigonometric barrier functions

A primal–dual interior-point method (IPM) based on a new class of proximity functions is proposed for solving Semidefinite Optimization (SDO) problems. The proposed functions are induced from the kernel functions with trigonometric barrier terms. We derive iteration complexity of large-update IPMs for SDO as O(nlognlognϵ). This improves the result obtained in Li and Zhang (2015) for linear optimization and matches to the bound for the so-called self-regular kernel functions.

A Parametric Kernel Function Yielding the Best Known Iteration Bound of Interior-Point Methods for Semidefinite Optimization

In this paper, a class of large-update primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function are presented. The proposed kernel function is not only used for determining the search directions but also for measuring the distance between the given iterate and the center for the algorithms. By means of the Nesterov and Todd scaling scheme, the currently best known iteration bounds for large-update methods is established.

Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term

In this paper, we present a class of large- and small-update primal-dual interior-point methods forsemidefinite optimization based on a parametric kernel function with a trigonometric barrier term.For both versions of the kernel-based interior-point methods, the worst case iteration bounds are established, namely,$O(n^{\frac{2}{3}}\log\frac{n}{\varepsilon})$ and $O(\sqrt{n}\log\frac{n}{\varepsilon})$, respectively.These results match the ones obtained in the linear optimization case.

Improved Complexity Analysis of Full Nesterov–Todd Step Interior-Point Methods for Semidefinite Optimization

In this paper, we present an improved convergence analysis of full Nesterov---Todd step feasible interior-point method for semidefinite optimization, and extend it to the infeasible case. This improvement due to a sharper quadratic convergence result, which generalizes a known result in linear optimization and leads to a slightly wider neighborhood for the iterates in the feasible algorithm and for the feasibility steps in the infeasible algorithm. For both versions of the full Nesterov---Todd step interior-point methods, we derive the same order of the iteration bounds as the ones obtained in linear optimization case.

A generic primal–dual interior-point method for semidefinite optimization based on a new class of kernel functions

In this paper, we present a class of polynomial-time primal–dual interior-point methods (IPMs) for semi-definite optimization based on a new class of kernel functions. This class is fairly general and includes the class of finite kernel functions [Y.Q. Bai, M. El Ghami and C. Roos, A new efficient large-update primal–dual interior-point method based on a finite barrier, SIAM J. Optim. 13(3) (2003), pp. 766–782]: the corresponding barrier functions have a finite value at the boundary of the feasible region. They are not exponentially convex and also not strongly convex like many usual barrier functions. We show that the IPMs based on these functions have favourable complexity results. To achieve this, several new tools are derived in the analysis. The kernel functions depend on parameters p∈[0, 1] and σ≥1. When those parameters are appropriately chosen, then the iteration bound of large-update IPMs based on these functions, coincide with the currently best known bounds for primal–dual IPMs.

A new full-Newton step O(n) infeasible interior-point algorithm for semidefinite optimization

Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed a primal-dual infeasible interior-point algorithm with the currently best iteration bound for linear optimization problems. Since the algorithm uses only full Newton steps, it has the advantage that no line-searches are needed. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of full-Newton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates either by finding an ε-solution or by detecting that the primal-dual problem pair has no optimal solution (X *,y *,S *) with vanishing duality gap such that the eigenvalues of X * and S * do not exceed ζ. The iteration bound coincides with the currently best iteration bound for semidefinite optimization problems.

New Complexity Analysis of the Primal–Dual Method for Semidefinite Optimization Based on the Nesterov–Todd Direction

Interior-point methods for semidefinite optimization have been studied intensively in recent times, due to their polynomial complexity and practical efficiency. In this paper, first we present some technical results about symmetric matrices. Then, we apply these results to give a unified analysis for both large update and small update interior-point methods for SDP based on the Nesterov–Todd (NT) direction.