The cubic algorithm

A nongradient algorithm for nonlinear nonconvex Lipschitzian optimization problems is proposed.     

The octahedral algorithm,a new simplicial fixed point algorithm

A new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented. It uses the restrart technique of Merrill to improve the accuracy of the solution. The algorithm is shown to converge quadratically under certain conditions. The algorithm should be efficient and relatively easy to implement.Partially supported by the Western Michigan University Sabbatical and Faculty Research Funds.     

The linear discrete Polya algorithm

We give a detailed analysis of the rate of convergence of the p^th power minimum of an affine subspace of as p → ∞.     

The bound improving sequence algorithm

The purpose of this paper is to report on a new tool to help solve 0–1 LP's. It consists of a sequence of bounds that, under proper conditions, bridge the duality gap, i.e., converges in a finite number of steps to the optimal value of the objective function of the problem studied. As a by-product an optimal solution for that problem is produced. Computational experience is reported.     

The tamed unadjusted Langevin algorithm

In this article, we consider the problem of sampling from a probability measure $\pi$ having a density on ${\mathbb{R}}^d$ proportional to $x?{\mathrm{e}}^{?U\left(x\right)}$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential $U$ is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in $V$-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and $\pi$, as well as weak error bounds. Numerical experiments are presented which support our findings.     

The Lascoux, Leclerc and Thibon algorithm and Soergel's tilting algorithm

We generalize Soergel's tilting algorithm to singular weights and deduce from this the validity of the Lascoux-Leclerc-Thibon conjecture on the connection between the canonical basis of the basic submodule of the Fock module and the representation theory of the Hecke-algebras at root of unity. Supported in part by Programa Reticulados y Ecuaciones and by FONDECYT grant 1051024.     

The disengagement algorithm or a new generalization of the exclusion algorithm

Let us consider a finite inf semilattice G with a set 1 of internal binary operations 1₁, isotono, satisfying certain conditions of no dispersion, of increasing and of substitution, and so that the greatest lower bound is distributive relatively to 1₁. A finite subset A of G being given, this article gives a method for enumerating the maximal elements of the sub-algebra A¹ generated by A with regard to 1, when A¹ is finite. This method, called disengagement algorithm, allows to examine each element once; it generalizes an algorithm giving the maximal n-rectangles of a part of a product of distributive lattices algorithm which already generalized a conjecture of Tison in Boolean algebra. Two applications are developed.     

The partial total least squares algorithm

In this paper, an improved algorithm PTLS for solving total least squares (TLS) problems AX ≈ B is presented. As only a basis of the right singular subspace associated with the smallest singular values of the data [A; B] is needed, the computational cost can be reduced considerably by using the partial SVD algorithm. This algorithm computes in an efficient way a basis for the left and/or right singular subspace of a matrix associated with its smallest singular values.An analysis of the operation counts, as well as computational results, show the relative efficiency of PTLS with respect to the classical TLS algorithm. Typically, PTLS reduces the computation time with a factor 2.     

The ellipsoid algorithm using parallel cuts

We present an ellipsoid algorithm using parallel cuts which is robust and conceptually simple. If the ratio fo the distance between the parallel cuts under consideration and the corresponding radius of the current ellipsoid is less than or equal to some constant, it is called the canonical case. Applying our algorithm to this case the volume of the next ellipsoid decreases by a factor which is, at worst, exp For the noncanonical case, we first add an extra constraint to make it a canonical case in a higher-dimensional space, then apply our algorithm to this canonical case, and finally reduce it back to the original space. Some interesting variants are also presented to show the flexibility of our basic algorithm.     

The Schur algorithm and its applications

The Schur algorithm and its time-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems. These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves. In this paper, the Schur and fast Chokesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.The work of this author was supported by the Exxon Education FoundationThe work of this author was supported by the Air Force Office of Scientific Research under Grant AFOSR-82-0135A.     

The extrapolated interval global optimization algorithm

This paper presents a new approach based on extrapolation to accelerate the linear convergence process of Vectorized Moore–Skelboe (VMS) algorithm. The VMS is a modified version of basic Moore–Skelboe (MS) algorithm, where the vectorization is used as a means to speed up the basic MS algorithm. We propose to further accelerate the converging process of VMS from linear to quadratic by combining the Richardson extrapolation technique with VMS. The effectiveness of the proposed algorithm is tested on various multivariate examples and compared with the unaccelerated conventional method, i.e., MS and well-known optimization software GlobSol. The test results show that the proposed extrapolation-based VMS offer considerable speed improvements over both the existing algorithms.     

The amortized complexity of Henriksen's algorithm

Henriksen's algorithm is a priority queue implementation that has been proposed for the event list found in discrete event simulations. It offers several practical advantages over heaps and tree structures.Although individual insertions ofO(n) complexity can easily be demonstrated to exist, the self-adjusting nature of the data structure seems to ensure that these will be rare. For this reason, a better measure of total running time is theamortized complexity: the worst case over a sequence of operations, rather than for a single operation.We show that Henriksen's algorithm has an amortized complexity of(n ^1/2) per insertion,O(1) per extract_min operation, andO(logn) for isolated deletions.     

The auction algorithm for the transportation problem

The auction algorithm is a parallel relaxation method for solving the classical assignment problem. It resembles a competitive bidding process whereby unassigned persons bid simultaneously for objects, thereby raising their prices. Once all bids are in, objects are awarded to the highest bidder. This paper generalizes the auction algorithm to solve linear transportation problems. The idea is to convert the transportation problem into an assignment problem, and then to modify the auction algorithm to exploit the special structure of this problem. Computational results show that this modified version of the auction algorithm is very efficient for certain types of transportation problems.     

The alternating basis algorithm for assignment problems

The purpose of this paper is to present a new primal extreme point algorithm for solving assignment problems which both circumvents and exploits degeneracy. The algorithm is based on the observation that the degeneracy difficulties of the simplex method result from the unnecessary inspection of alternative basis representations of the extreme points. This paper characterizes a subsetQ of all bases that are capable of leading to an optimal solution to the problem if one exists. Using this characterization, an extreme point algorithm is developed which considers only those bases inQ. Computational results disclose that the new algorithm is substantially more efficient than previously developed primal and primal-dual extreme point (simplex) methods for assignment problems.