Yanqin Bai

A Comparative Study of Kernel Functions for Primal-Dual Interior-Point Algorithms in Linear Optimization

Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on these kernel functions. In both the algorithm and its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of kernel function in the new class. We apply this scheme to seven specific kernel functions. Some of these functions are self-regular, and others are not. One of the functions differs from the others, and from all self-regular functions, in the sense that its growth term is linear. Iteration bounds for both large- and small-update methods are derived. It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely,

A New Efficient Large-Update Primal-Dual Interior-Point Method Based on a Finite Barrier

O(\sqrt{n}\log\frac{n}{\e})

Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

. For large-update methods the best obtained bound is

A polynomial-time algorithm for linear optimization based on a new simple kernel function

O(\sqrt{n}(\log n)\log\frac{n}{\e})

A primal‐dual interior-point method for linear optimization based on a new proximity function

, which until now has been the best known bound for such methods.

Optimal trade-off portfolio selection between total risk and maximum relative marginal risk

We introduce a new barrier-type function which is not a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, the iteration bound of a large-update interior-point method based on this function is shown to be

A full-step interior-point algorithm for second-order cone optimization based on a simple locally kernel function

O({\sqrt{n}\,({\rm log}\,n)\,{\rm log}\,\frac{n}{\varepsilon}})

Portfolio selection with the effect of systematic risk diversification: formulation and accelerated gradient algorithm

, which is as good as the currently best known bound for large-update methods. The recently introduced property of \emph{exponential convexity} for the kernel function underlying the barrier function, as well as the strong convexity of the kernel function, are crucial in the analysis.

Splitting augmented Lagrangian method for optimization problems with a cardinality constraint and semicontinuous variables

AbstractInterior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J. Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximities for linear optimization (LO) problems. They also extended the approach for LO to SDO. In this paper we present a primal-dual interior-point algorithm for SDO problems based on a simple kernel function which was first presented at the Proceedings of Industrial Symposium and Optimization Day, Australia, November 2002; the function is not self-regular. We derive the complexity analysis for algorithms based on this kernel function, both with large- and small-updates. The complexity bounds are

Exploiting Group Symmetry in Truss Topology Optimization

\mathrm{O}(qn)\log\frac{n}{\epsilon}