Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.     

Hybrid Flow-Shop: a Memetic Algorithm Using Constraint-Based Scheduling for Efficient Search

The paper considers the hybrid flow-shop scheduling problem with multiprocessor tasks. Motivated by the computational complexity of the problem, we propose a memetic algorithm for this problem in the paper. We first describe the implementation details of a genetic algorithm, which is used in the memetic algorithm. We then propose a constraint programming based branch-and-bound algorithm to be employed as the local search engine of the memetic algorithm. Next, we present the new memetic algorithm. We lastly explain the computational experiments carried out to evaluate the performance of three algorithms (genetic algorithm, constraint programming based branch-and-bound algorithm, and memetic algorithm) in terms of both the quality of the solutions produced and the efficiency. These results demonstrate that the memetic algorithm produces better quality solutions and that it is very efficient.     

A simulated annealing driven multi-start algorithm for bound constrained global optimization

A derivative-free simulated annealing driven multi-start algorithm for continuous global optimization is presented. We first propose a trial point generation scheme in continuous simulated annealing which eliminates the need for the gradient-based trial point generation. We then suitably embed the multi-start procedure within the simulated annealing algorithm. We modify the derivative-free pattern search method and use it as the local search in the multi-start procedure. We study the convergence properties of the algorithm and test its performance on a set of 50 problems. Numerical results are presented which show the robustness of the algorithm. Numerical comparisons with a gradient-based simulated annealing algorithm and three population-based global optimization algorithms show that the new algorithm could offer a reasonable alternative to many currently available global optimization algorithms, specially for problems requiring ‘direct search’ type algorithm.     

An alogrithm for monotonic global optimization problems ∗

We propose an algorithm to locate a global maximum of an increasing function subject to an increasing constraint on the cone of vectors with nonnegative coordinates. The algorithm is based on the outer approximation of the feasible set. We eastablish the con vergence of the algorithm and provide a number of numerical experiments. We also discuss the types of constraints and objective functions for which the algorithm is best suited     

Augmented Lagrangian algorithms for linear programming

We introduce new augmented Lagrangian algorithms for linear programming which provide faster global convergence rates than the augmented algorithm of Polyak and Treti'akov. Our algorithm shares the same properties as the Polyak-Treti'akov algorithm in that it terminates in finitely many iterations and obtains both primal and dual optimal solutions. We present an implementable version of the algorithm which requires only approximate minimization at each iteration. We provide a global convergence rate for this version of the algorithm and show that the primal and dual points generated by the algorithm converge to the primal and dual optimal set, respectively.     

Fractions continues sur les surfaces de Veech

We define a geometrical continued fraction algorithm in the setting of regular polygons with an even number of sides., The definition of the algorithm uses linear transformations generating a group conjugated to an index 2 subgroup of a Hecke group. We give Markov conditions allowing the iteration of the algorithm. We compute the natural extension and the invariant measure for each of the additive and multiplicative versions of this algorithm.      

A fast computational algorithm for the Legendre-Fenchel transform

We investigate a fast algorithm, introduced by Brenier, which computes the Legendre-Fenchel transform of a real-valued function. We generalize his work to boxed domains and introduce a parameter in order to build an iterative algorithm. The new approach of separating primal and dual spaces allows a clearer understanding of the algorithm and yields better numerical behavior. We extend known complexity results and give new ones about the convergence of the algorithm.     

An Algorithm for Global Optimization using the Taylor–Bernstein Form as Inclusion Function

We investigate the use of higher order inclusion functions in the Moore–Skelboe (MS) algorithm of interval analysis (IA) for unconstrained global optimization. We first propose an improvement of the Taylor–Bernstein (TB) form given in (Lin and Rokne (1996) 101) which has the property of higher order convergence. We make the improvement so that the TB form is more effective in practice. We then use the improved TB form as an inclusion function in a prototype MS algorithm and also modify the cut-off test and termination condition in the algorithm. We test and compare on several examples the performances of the proposed algorithm, the MS algorithm, and the MS algorithm with the Taylor model of Berz and Hoffstatter (1998; 97) as inclusion function. The results of these (preliminary) tests indicate that the proposed algorithm with the improved TB form as inclusion function is quite effective for low to medium dimension problems studied.     

An Extension of Least Angle Regression Based on the Information Geometry of Dually Flat Spaces

We extend the least angle regression algorithm using the information geometry of dually flat spaces. The extended least angle regression algorithm is used for estimating parameters in generalized linear regression, and it can be also used for selecting explanatory variables. We use the fact that a model manifold of an exponential family is a dually flat space. In estimating parameters, curves corresponding to bisectors in the Euclidean space play an important role. Originally, the least angle regression algorithm is used for estimating parameters and selecting explanatory variables in linear regression. It is an efficient algorithm in the sense that the number of iterations is the same as the number of explanatory variables. We extend the algorithm while keeping this efficiency. However, the extended least angle regression algorithm differs significantly from the original algorithm. The extended least angle regression algorithm reduces one explanatory variable in each iteration while the original algorithm increases one explanatory variable in each iteration. We show results of the extended least angle regression algorithm for two types of datasets. The behavior of the extended least angle regression algorithm is shown. Especially, estimates of parameters become smaller and smaller, and vanish in turn.     

Prime witnesses in the Shor algorithm and the Miller-Rabin algorithm

We prove that prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm which satisfy the condition of Fermat’s little theorem. We describe the set of natural numbers, whose prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm. We find all such numbers less than 100,000,000 and experimentally study the rate of increase of the ratio of the quantity of such numbers to the quantity of Carmichael numbers.     

Global optimization of robust chance constrained problems

We propose a stochastic algorithm for the global optimization of chance constrained problems. We assume that the probability measure with which the constraints are evaluated is known only through its moments. The algorithm proceeds in two phases. In the first phase the probability distribution is (coarsely) discretized and solved to global optimality using a stochastic algorithm. We only assume that the stochastic algorithm exhibits a weak* convergence to a probability measure assigning all its mass to the discretized problem. A diffusion process is derived that has this convergence property. In the second phase, the discretization is improved by solving another nonlinear programming problem. It is shown that the algorithm converges to the solution of the original problem. We discuss the numerical performance of the algorithm and its application to process design.     

A simplification of the double-sweep algorithm to solve the k-shortest path problem

In this paper, we define the k-shortest path problem, which will be used to model the problem of routing aircraft through a network of airfields. This problem finds the optimal alternative routes from one or more origins to one or more destinations. We solve this problem using the double-sweep algorithm. We present a simplification to the double-sweep algorithm, and show that this simplification reduces the computational complexity of the algorithm by a factor of k. We prove that the simplified double-sweep algorithm converges. Finally, we demonstrate this algorithm on a small airlift network.     

Computing absorption probabilities for a Markov chain

We present a new matrix construction algorithm for computing absorption probabilities for a finite, reducible Markov chain. The construction algorithm contains two steps: matrix augmentation and matrix reduction. The algorithm requires more memory and less execution time than the calculation of absorption probabilities by the LU decomposition. We apply the algorithm to a Markovian model of a production line.     

On the optimal design of the randomized unbiased Monte Carlo estimators

We consider a class of unbiased Monte Carlo estimators and develop an efficient algorithm to produce the distribution of the optimal random truncation level. We establish the convergence and optimality results of the associated algorithm and also derive its exact complexity. We find this algorithm has a much lower complexity as compared to the existing one in the literature. The proposed algorithm is also applicable to optimization problems arising in supply chain management, such as economic reorder interval problem.     

Bounded delay packet scheduling in a bounded buffer

We study buffer management in QoS-enabled network switches in the bounded delay model where each packet has a weight and a deadline. We consider the more realistic situation where the switch has a finite buffer size. A 9.82-competitive algorithm is known for the case of multiple buffers. Recently, for the single buffer case, a 3-competitive deterministic algorithm and a 2.618-competitive randomized algorithm were found. We give a simple deterministic 2-competitive algorithm for the single buffer case.     

A direct algorithm to construct the minimal Z-pairs for rational functions

In this paper, we present a direct algorithm to construct the minimal Z-pairs for rational functions. We describe a Maple implementation of the algorithm and show timing comparisons between this algorithm and other related algorithms. We also summarize an analogous algorithm for the q-difference case.     

A primal—dual affine-scaling potential-reduction algorithm for linear programming

We propose a potential-reduction algorithm which always uses the primal—dual affine-scaling direction as a search direction. We choose a step size at each iteration of the algorithm such that the potential function does not increase, so that we can take a longer step size than the minimizing point of the potential function. We show that the algorithm is polynomial-time bounded. We also propose a low-complexity algorithm, in which the centering direction is used whenever an iterate is far from the path of centers.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.     

An objective space cut and bound algorithm for convex multiplicative programmes

Multiplicative programming problems are global optimisation problems known to be NP-hard. In this paper we propose an objective space cut and bound algorithm for approximately solving convex multiplicative programming problems. This method is based on an objective space approximation algorithm for convex multi-objective programming problems. We show that this multi-objective optimisation algorithm can be changed into a cut and bound algorithm to solve convex multiplicative programming problems. We use an illustrative example to demonstrate the working of the algorithm. Computational experiments illustrate the superior performance of our algorithm compared to other methods from the literature.     

Structure Algorithms for Partially Ordered Isotonic Regression

Abstract

An algorithm for isotonic regression is called a structure algorithm if it searches for a “solution partition”—that is, a class of sets on each of which the isotonic regression is a constant. We discuss structure algorithms for partially ordered isotonic regression. In this article we provide a new class of structure algorithms called the isotonic block class (IBC) type algorithms. One of these is called the isotonic block class with recursion method (IBCR) algorithm, which works for partially ordered isotonic regression. It is a generalization of the pooled adjacent violators algorithm and is simpler than the min-max algorithm. We also give a polynomial time algorithm—the isotonic block class with stratification (IBCS) algorithm for matrix-ordered isotonic regression. We demonstrate the efficiency of our IBCR algorithm by using simulation to estimate the relative frequencies of the numbers of level sets of isotonic regressions on certain two-dimensional grids with the matrix order.     

Symmetric orthogonal approximation to symmetric tensors with applications to image reconstruction

The main objective of this paper is to study an approximation of symmetric tensors by symmetric orthogonal decomposition. We propose and study an iterative algorithm to determine a symmetric orthogonal approximation and analyze the convergence of the proposed algorithm. Numerical examples are reported to demonstrate the effectiveness of the proposed algorithm. We also apply the proposed algorithm to represent correlated face images. We demonstrate better face image reconstruction results by combining principal components and symmetric orthogonal approximation instead of combining principal components and higher‐order SVD results.