Interior-point Algorithm For Semidefinite Optimization Research Articles

Complexity analysis and numerical implementation of a new interior-point algorithm for semidefinite optimization

We generalize Zhang and Xu's (2011) [22] interior point algorithm for linear optimization to semidefinite optimization problems in order to define a new search direction. The symmetrization of the search direction is based on the full Nesterov-Todd scaling scheme. Moreover, we show that the obtained algorithm solves the studied problem in polynomial time and that the short-step algorithm has the best-known iteration bound, namely O(nlog⁡nε)-iterations. Finally, we report a comparative numerical study to show the efficiency of our proposed algorithm.

A wide neighbourhood primal-dual second-order corrector interior point algorithm for semidefinite optimization

In this paper, we proposed a new primal-dual second-order corrector interior-point algorithm for semidefinite optimization. The algorithm is based on Darvay–Takács neighbourhood of the central path. In the new algorithm, the search directions are determined by the Darvay–Takács's direction and a second-order corrector direction in each iteration. The iteration complexity bound is for the Nesterov–Todd scaling direction, which coincides with the best-known complexity results for semidefinite optimization. Finally, numerical experiments show that the proposed algorithm is promising.

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.

An predictor–corrector interior-point algorithm for semidefinite optimization based on a wide neighbourhood

In this paper, we propose a new predictor–corrector interior-point algorithm for semidefinite optimization based on a wide neighbourhood of the central path. We show that, in addition to the predictor step, each corrector step decreases the duality gap as well. We also prove that the iteration complexity of the proposed algorithm coincides with the best iteration bound for small neighbourhood algorithms that use the Nesterov–Todd direction. Finally, some numerical results are provided as well.

A new wide neighbourhood primal-dual interior-point algorithm for semidefinite optimization

ABSTRACTIn this paper, we present a wide neighbourhood primal-dual interior-point algorithm for semidefinite optimization. We define a new wide neighbourhood , which is an extension of the Darvay-Takács's wide neighbourhood for linear optimization to semidefinite optimization. We use an algebraic reformulation of the centering equation along with Zhang's symmetrization operator to create symmetric search directions. After applying Newton's method to the resulting system, we recommend using the square root function for simplicity. We consider the Newton direction as the sum of two other directions, corresponding to the negative and positive parts of the right-hand side of the centring equation. Starting with a feasible point , we prove that our algorithm has complexity the same as small neighbourhood path-following algorithms for semidefinite optimization that use the Nesterov-Todd directions. To the best of our knowledge, this is the first large neighbourhood path-following interior-point algorithm that uses a different neighbourhood from those available for semidefinite optimization.

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 full Nesterov–Todd step infeasible interior-point algorithm for semidefinite optimization based on a kernel function

In this paper, we propose a new complexity analysis of the full Nesterov–Todd step infeasible interior-point algorithm for semidefinite optimization. With a specific feasibility step and the centering step induced by a well-known kernel function, the property of exponential convexity of the kernel function underlying the matrix barrier function is crucial in the analysis and enable us easily to estimate the proximity of iterates to center path. The analysis of the algorithm is simplified and the iteration bound obtained coincides with the currently best iteration bound for infeasible interior-point algorithm.

On the extension of an arc-search interior-point algorithm for semidefinite optimization

This paper concerns an extension of the arc-search strategy that was proposed by Yang [26] for linear optimization to semidefinite optimization case. Based on the Nesterov-Todd direction as Newton search direction it is shown that the complexity bound of the proposed algorithm is of the same order as that of the corresponding algorithm for linear optimization. Some preliminary numerical results indicate that our primal-dual arc-search path-following method is promising for solving the semidefinite optimization problems.

An interior-point algorithm for semidefinite optimization based on a new parametric kernel function

An interior-point algorithm for semidefinite optimization based on a new parametric kernel function

A unified complexity analysis of interior point methods for semidefinite problems based on trigonometric kernel functions

In this paper, we present an interior-point algorithm for semidefinite optimization (SDO) problems based on a new generic trigonometric kernel function, which is constructed by introducing some new conditions on the kernel function. Based on these conditions, we propose a new trigonometric kernel function and present some properties of this function. We present some complexity results for the generic kernel function and prove that, a large-update primal–dual interior-point method for solving SDO problems with this new kernel function enjoys as worst case iteration complexity bound which matches the currently best known complexity bound for large update methods. Moreover, some numerical results show that the new proposed kernel function has better results than the other trigonometric kernel functions.

A Wide Neighborhood Second-order Predictor-corrector Interior-point Algorithm for Semidefinite Optimization with Modified Corrector Directions

A Wide Neighborhood Second-order Predictor-corrector Interior-point Algorithm for Semidefinite Optimization with Modified Corrector Directions

A full NT-step infeasible interior-point algorithm for semidefinite optimization

In this paper, a full Nesterov–Todd-step infeasible interior-point algorithm is presented for semidefinite optimization (SDO) problems. In contrast of some classical interior-point algorithms for SDO problems, this algorithm does not need to perform computationally expensive calculations for centering steps which are needed for classical interior-point methods. The convergence analysis of the algorithm is shown and it is also proved that the complexity bound of the algorithm coincides with the currently best iteration bound obtained by infeasible interior-point algorithms for this class of optimization problems.

A Mehrotra predictor-corrector interior-point algorithm for semidefinite optimization

This paper proposes a second-order Mehrotra-type predictor-corrector feasible interior-point algorithm for semidefinite optimization problems. In each iteration, the algorithm computes the Newton search directions through a new form of combination of the predictor and corrector directions. Using the Ai-Zhang wide neighborhood for linear complementarity problems, it is shown that the complexity bound of the algorithm is $O(\sqrt{n}\log \varepsilon^{-1})$ for the Nesterov-Todd search direction and $O({n}\log \varepsilon^{-1})$ for the Helmberg-Kojima-Monteiro search directions.

An $$O\left(\sqrt{n}L\right)$$ O n L wide neighborhood interior-point algorithm for semidefinite optimization

In this paper, we propose a primal–dual interior-point method for semidefinite optimization problems. The algorithm is based on a new class of search directions and the Ai-Zhang’s wide neighborhood for monotone linear complementarity problems. The theoretical complexity of the new algorithm is calculated. It is investigated that the proposed algorithm has polynomial iteration complexity \(O(\sqrt{n}L)\) and coincides with the best known iteration bound for semidefinite optimization problems.

An Adaptive Infeasible Interior-Point Algorithm with Full Nesterov-Todd Step for Semidefinite Optimization

We present a new full Nesterov and Todd step infeasible interior-point algorithm for semi-definite optimization. The algorithm decreases the duality gap and the feasibility residuals at the same rate. In the algorithm, we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Every main iteration of the algorithm consists of a feasibility step and some centering steps. We show that the algorithm converges and finds an approximate solution in polynomial time. A numerical study is made for the numerical performance. Finally, a comparison of the obtained results with those by other existing algorithms is made.

A primal-dual large-update interior-point algorithm for semi-definite optimization based on a new kernel function

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.

A primal-dual large-update interior-point algorithm for semi-definite optimization based on a new kernel function

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.

Simplified analysis for full-Newton step infeasible interior-point algorithm for semidefinite programming

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.

Kernel function based interior-point algorithms for semidefinite optimization

We propose a primal-dual interior-point algorithm for semidefinite optimization(SDO) based on a class of kernel functions which are both eligible and self-regular. New search direc- tions and proximity measures are defined based on these functions. We show that the algorithm has O( √ nlog n ) and O( √ nlognlog n ) complexity results for small- and large-update meth- ods, respectively. These are the best known complexity results for such methods. This is the first algorithm for SDO based on this kernel function, as far as we know.

A full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular proximity

A full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular proximity