Takashi Tsuchiya

Polynomiality of primal-dual algorithms for semidefinite linear complementarity problems based on the Kojima-Shindoh-Hara family of directions

Kojima, Shindoh and Hara proposed a family of search directions for the semidefinite linear complementarity problem (SDLCP) and established polynomial convergence of a feasible short-step path-following algorithm based on a particular direction of their family. The question of whether polynomiality could be established for any direction of their family thus remained an open problem. This paper answers this question in the affirmative by establishing the polynomiality of primal-dual interior-point algorithms for SDLCP based on any direction of the Kojima, Shindoh and Hara family of search directions. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SDLCP.

A strong bound on the integral of the central path curvature and its relationship with the iteration-complexity of primal-dual path-following LP algorithms

The main goals of this paper are to: i) relate two iteration-complexity bounds derived for the Mizuno-Todd-Ye predictor-corrector (MTY P-C) algorithm for linear programming (LP), and; ii) study the geometrical structure of the LP central path. The first iteration-complexity bound for the MTY P-C algorithm considered in this paper is expressed in terms of the integral of a certain curvature function over the traversed portion of the central path. The second iteration-complexity bound, derived recently by the authors using the notion of crossover events introduced by Vavasis and Ye, is expressed in terms of a scale-invariant condition number associated with m × n constraint matrix of the LP. In this paper, we establish a relationship between these bounds by showing that the first one can be majorized by the second one. We also establish a geometric result about the central path which gives a rigorous justification based on the curvature of the central path of a claim made by Vavasis and Ye, in view of the behavior of their layered least squares path following LP method, that the central path consists of

Numerical experiments with universal barrier functions for cones of Chebyshev systems

{\mathcal{O}}(n^2)

Convergence analysis of the projective scaling algorithm based on a long-step homogeneous affine scaling algorithm

long but straight continuous parts while the remaining curved part is relatively “short”.

An extension of Chubanov's polynomial-time linear programming algorithm to second-order cone programming

We develop an information geometric approach to conic programming. Information geometry is a differential geometric framework specifically tailored to deal with convexity, naturally arising in information science including statistics, machine learning and signal processing etc. First we introduce an information geometric framework of conic programming. Then we focus on semidefinite and symmetric cone programs. Recently, we demonstrated that the number of iterations of Mizuno–Todd–Ye predictor–corrector primal–dual interior-point methods is (asymptotically) expressed with an integral over the central trajectory called “the curvature integral”. The number of iterations of the algorithm is approximated surprisingly well with the integral even for fairly large linear/semidefinite programs with thousands of variables. Here we prove that “the curvature integral” admits a rigorous differential geometric expression based on information geometry. We also obtain an interesting information geometric global theorem on the central trajectory for linear programs. Together with the numerical evidence in the aforementioned work, we claim that “the number of iterations of the interior-point algorithm is expressed as a differential geometric quantity.”

Weak Infeasibility in Second Order Cone Programming

Abstract Based on previous explicit computations of universal barrier functions, we describe numerical experiments for solving certain classes of convex optimization problems. The comparison is given of the performance of the classical affine-scaling algorithm with the similar algorithm built upon the universal barrier function.

Curvature integrals and iteration complexities in SDP and symmetric cone programs

In order to classify fully developed chaos produced by simple one-dimensional maps the Pearson walk which is a special kind of two-dimensional random walk is introduced to visualize, in principle, all the orbits starting from different initial points in one diagram. Chaos of the tent map, standard baker transformation and the logistic map is clearly distinguished from one another in our Pearson-walk representation, which also provides information on not-fully developed chaos that emerges for smaller parameter values for each map. In the present visualization it is also clearly seen how characteristics of the so-called window is differentiated from those of the purely periodic region in the logistic map.

Proximity of Weighted and Layered Least Squares Solutions

In this paper we give a new convergence analysis of a projective scaling algorithm. We consider a long-step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem. This algorithm takes a fixed fraction λ≤2/3 of the way towards the boundary of the nonnegative orthant at each iteration. The iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection, which generates the same search direction as the original projective scaling algorithm at each iterate. The recent convergence results for the long-step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm. Specifically, we will show (i) polynomiality of the algorithm with complexities of O(nL) and O(n2L) iterations for λ<2/3 and λ=2/3, respectively; (ii) global covnergence of the algorithm when the optimal face is unbounded; (iii) convergence of the primal iterates to a relative interior point of the optimal face; (iv) convergence of the dual estimates to the analytic center of the dual optimal face; and (v) convergence of the reduction rate of the objective function value to 1−λ.