The Complexity of Self-Regular Proximity Based Infeasible IPMs

Primal-Dual Interior-Point Methods (IPMs) have shown their power in solving large classes of optimization problems. In this paper a self-regular proximity based Infeasible Interior Point Method (IIPM) is proposed for linear optimization problems. First we mention some interesting properties of a specific self-regular proximity function, studied recently by Peng and Terlaky, and use it to define infeasible neighborhoods. These simple but interesting properties of the proximity function indicate that, when the current iterate is in a large neighborhood of the central path, large-update IIPMs emerge as the only natural choice. Then, we apply these results to design a specific self-regularity based dynamic large-update IIPM in large neighborhood. The new dynamic IIPM always takes large-updates and does not utilize any inner iteration to get centered. An worst-case iteration bound of the algorithm is established. Finally, we report the main results of our computational experiments.     

Exact two steps SOCP/SDP formulation for a modified conic trust region subproblem

In this paper, we study a modified version of the conic trust region subproblem which arises within a class of nonlinear programming algorithms. First using a variant of S-Lemma, we give an SOCP/SDP formulation which gives its optimal objective value. Then using the parametrization approach of Dinkelbach and the known exact SOCP/SDP relaxation of the extended trust region subproblem, we find its optimal solution. Finally, some preliminary numerical results are given.     

Regularization tools and robust optimization for ill-conditioned least squares problem: A computational comparison

Least squares problems arise frequently in many disciplines such as image restorations. In these areas, for the given least squares problem, usually the coefficient matrix is ill-conditioned. Thus if the problem data are available with certain error, then after solving least squares problem with classical approaches we might end up with a meaningless solution. Tikhonov regularization, is one of the most widely used approaches to deal with such situations. In this paper, first we briefly describe these approaches, then the robust optimization framework which includes the errors in problem data is presented. Finally, our computational experiments on several ill-conditioned standard test problems using the regularization tools, a Matlab package for least squares problem, and the robust optimization framework, show that the latter approach may be the right choice.     

On self-regular IPMs

Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best known theoretical worst-case iteration bound, but work very poorly in practice. To the contrary, the so-called large-update IPMs have superior practical performance but with relatively weaker theoretical results. In this paper we discuss the new algorithmic variants and improved complexity results with respect to the new family of Self-Regular proximity based IPMs for Linear Optimization problems, and their generalizations to Conic and Semidefinite Optimization This research was supported by the MITACS project “New IPMs and Software for Convex Conic-Linear Optimization and Their Application to Solve VLSI Circuit Layout Problems”, by an NSERC discovery grant, and the CRC program. The first author would also like to thank the Iranian Ministry of Science, Research and Technology for supporting his research.     

An efficient,one-pot,three-component procedure for the synthesis of chiral spirooxindolopyrrolizidines via catalytic highly enantioselective 1,3-dipolar cycloaddition

The catalytic, highly regio-, diastereo-, and enantioselective synthesis of a small library of chiral spirooxindolopyrrolizidines via a three-component 1,3-dipolar cycloaddition reaction of azomethine ylides, derived from isatin, with electron-deficient dipolarophiles, 3-(2-alkenoyl)-1,3-oxazolidin-2-ones, is described. A chiral copper(II) complex of cyclohexane-1,2-bis(arylmethyleneamine) catalyzed this process at room temperature. The reaction mechanism is discussed on the basis of the assignment of the absolute configuration of the cycloadducts.