A class of polynomial primal-dual interior-point algorithms for semidefinite optimization

In the present paper we present a class of polynomial primal-dual interior-point algorithms for semidefmite 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 methodswere obtained, namely,O（qn^（p＋q/q（P＋1）log n/ε and O（q^2√n）log n/ε,respectlvely.

Complexity analysis of interior-point algorithm based on a new kernel function for semidefinite optimization

Interior-point methods （IPMs） for linear optimization （LO） and semidefinite optimization （SDO） have become a hot area in mathematical programming in the last decades. In this paper, a new kernel function with simple algebraic expression is proposed. Based on this kernel function, a primal-dual interior-point methods （IPMs） for semidefinite optimization （SDO） is designed. And the iteration complexity of the algorithm as O（n^3/4 log n/ε） with large-updates is established. The resulting bound is better than the classical kernel function, with its iteration complexity O（n log n/ε） in large-updates case.

Complexity Analysis of an Interior Point Algorithm for the Semidefinite Optimization Based on a Kernel Function with a Double Barrier Term

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization（SDO） problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 〉 q2 〉 1, the algorithm has O（（q1 ＋ 1） nq1＋1/2（q1-q2）logn/ε）and O（（q1 ＋ 1）2（q1-q2）^3q1-2q2＋1√n logn/c） complexity results for large- and small-update methods, respectively.

A Full-Newton Step Feasible IPM for Semidefinite Optimization Based on a Kernel Function with Linear Growth Term

In this paper,we propose and analyze a full-Newton step feasible interior-point algorithm for semidefinite optimization based on a kernel function with linear growth term.The kernel function is used both for determining the search directions and for measuring the distance between the given iterate and theμ-center for the algorithm.By developing a new norm-based proximity measure and some technical results,we derive the iteration bound that coincides with the currently best known iteration bound for the algorithm with small-update method.In our knowledge,this result is the first instance of full-Newton step feasible interior-point method for SDO which involving the kernel function.

An Efficient Parameterized Logarithmic Kernel Function for Semidefinite Optimization

In this paper,we present a primal-dual interior point algorithm for semidefinite optimization problems based on a new class of kernel functions.These functions constitute a combination of the classic kernel function and a barrier term.We derive the complexity bounds for large and small-update methods respectively.We show that the best result of iteration bounds for large and small-update methods can be achieved,namely O(q√n(log√n)^q+1/q logn/ε)for large-update methods and O(q^3/2(log√q)^q+1/q√nlogn/ε)for small-update methods.We test the efficiency and the validity of our algorithm by running some computational tests,then we compare our numerical results with results obtained by algorithms based on different kernel functions.

A Wide Neighborhood Interior-Point Algorithm for Convex Quadratic Semidefinite Optimization

In this paper,we propose an interior-point algorithm based on a wide neighborhood for convex quadratic semidefinite optimization problems.Using the Nesterov–Todd direction as the search direction,we prove the convergence analysis and obtain the polynomial complexity bound of the proposed algorithm.Although the algorithm belongs to the class of large-step interior-point algorithms,its complexity coincides with the best iteration bound for short-step interior-point algorithms.The algorithm is also implemented to demonstrate that it is efficient.

Reduction of truss topology optimization

A reduction of truss topology design problem formulated by semidefinite optimization （SDO） is considered. The finite groups and their representations are introduced to reduce the stiffness and mass matrices of truss in size. Numerical results are given for both the original problem and the reduced problem to make a comparison.

A New Second-Order Mehrotra-Type Predictor-Corrector Algorithm for SDO

In Zhang’s recent works,a second-order Mehrotra-type predictor-corrector algorithm for linear optimization was extended to semidefinite optimization and derived that the algorithm for semidefinite optimization had3/2 0 T 0O（nlog（X）gS/e）iteration complexity based on the NT direction as Newton search direction.In this paper,we extend the second-order Mehrotra-type predictor-corrector algorithm for linear optimization to semidefinite optimization and discuss the polynomial convergence of the algorithm by modifying the corrector direction and new iterates.It is proved that the iteration complexity is reduced to0 0O（nlog XgS/e）,which coincides with the currently best iteration bound of Mehrotra-type predictor-corrector algorithm for semidefinite optimization.