Self-regular Function Research Articles

A Predictor-Corrector Algorithm for P ∗(κ)-Linear Complementarity Problems Based on a Specific Self-Regular Proximity Function

First-order predictor-corrector methods working in a large neighborhood of the central path are among the most efficient interior point methods. In Peng et al. (SIAM J. Optim. 15(4):1105–1127, 2005), based on a specific proximity function, a wide neighborhood of the central path is defined which matches the standard large neighborhood defined by the infinity norm. In this paper, we extend the predictor-corrector algorithm proposed for linear optimization in Peng et al. (SIAM J. Optim. 15(4):1105–1127, 2005) to P ∗(κ)-linear complementarity problems. Our algorithm performs two kinds of steps. In corrector steps, we use the specific self-regular proximity function to compute the search directions. The predictor step is the same as the predictor step of standard predictor-corrector method in the interior point method literature. We prove that our predictor-corrector algorithm has an \({\mathcal {O}}\left ((1+2\kappa )\sqrt {n}\log n\log \frac {(x^{0})^{T} s^{0}}{\epsilon }\right )\) iteration bound, which is the best known iteration complexity for problems of this type.

Kernel function-based primal-dual interior-point methods for symmetric cones optimization

In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization (SCO) based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity \(O(\sqrt r (\log r)^2 \log (r/\varepsilon ))\), which is as good as the convex quadratic semi-definite optimization analogue.

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.

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

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be \(O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)\). This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.

A FULL NT-STEP INFEASIBLE INTERIOR-POINT ALGORITHM FOR SEMIDEFINITE OPTIMIZATION BASED ON A SELF-REGULAR PROXIMITY

AbstractWe introduce a full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular function to provide the feasibility step and to measure proximity to the central path. The result of polynomial complexity coincides with the best known iteration bound for infeasible interior-point methods.

A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization

In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the classical logarithmic function, the prototype self-regular function, and non-self-regular kernel functions as special cases. The analysis of the algorithms in the paper follows the same line of arguments as in Bai et al. (SIAM J. Optim. 15:101–128, [2004]), where a variety of non-self-regular kernel functions were considered including the ones with linear and quadratic growth terms. However, the important case when the growth term is between linear and quadratic was not considered. The goal of this paper is to introduce such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. They match the currently best known iteration bounds for the prototype self-regular function with quadratic growth term, the simple non-self-regular function with linear growth term, and the classical logarithmic kernel function. In order to achieve these complexity results, several new arguments had to be used.

New complexity analysis of IIPMs for linear optimization based on a specific self-regular function

Primal–dual interior-point methods (IPMs) have shown their ability in solving large classes of optimization problems efficiently. Feasible IPMs require a strictly feasible starting point to generate the iterates that converge to an optimal solution. The self-dual embedding model provides an elegant solution to this problem with the cost of slightly increasing the size of the problem. On the other hand, infeasible interior point methods (IIPMs) can be initiated by any positive vector, and thus are popular in IPMs based software packages. In this paper we propose an adaptive large-update IIPM based on a specific self-regular proximity function, with barrier degree 1 + log n , that operates in a negative infinity neighborhood of the central path. An O ( n 3 2 log n log n ϵ ) worst-case iteration bound of the new algorithm is established. This iteration bound improves the so far best O ( n 2 log n ϵ ) iterations bound of IIPMs in a large neighborhood of the central path.

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 semidefinite 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 methods were obtained, namely, O(qn(p+q)/q(p+1)) log n/ɛ and O(q2 √n) log n/ɛ, respectively.

A dynamic large-update primal‐dual interior-point method for linear optimization

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, while the so-called large-update IPMs have superior practical performance but with relatively weaker theoretical results. In this article, by restricting us to linear optimization (LO), we first exploit some interesting properties of a self-regular proximity function, proposed recently by the authors of this work and Roos, and use it to define a neighborhood of the central path. These simple but interesting properties of the proximity function indicate that, when the current iterate is in a large neighborhood of the central path, then the large-update IPM emerges to be the only natural choice. Then, we apply these results to design a specific self-regularity based IPM. Among others, we show that this self-regularity based IPM can also predict precisely the change of the duality gap as the standard IPM does. Therefore, we can directly apply the modified IPM to the simplified self-dual homogeneous model for LO. This provides a remedy for an implementation issue of the new self-regular IPMs. A dynamic large-update IPM in large neighborhood is proposed. Different from traditional large-update IPMs, the new dynamic IPM always takes large-updates and does not utilize any inner iteration to get centered. An worst-case iteration bound of the algorithm is established.