Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm

Several numerical methods for solving nonlinear systems of equations assume that derivative information is available. Furthermore, these approaches usually do not consider the problem of finding all solutions to a nonlinear system. Rather, most methods output a single solution. In this paper, we address the problem of finding all roots of a system of equations. Our method makes use of a biased random-key genetic algorithm (BRKGA). Given a nonlinear system, we construct a corresponding optimization problem, which we solve multiple times, making use of a BRKGA, with areas of repulsion around roots that have already been found. The heuristic makes no use of derivative information. We illustrate the approach on seven nonlinear equations systems with multiple roots from the literature.     

Nonlinear Dynamical Systems and Adaptive Filters in Biomedicine

In this paper we present the application of a method of adaptive estimation using an algebra–geometric approach, to the study of dynamic processes in the brain. It is assumed that the brain dynamic processes can be described by nonlinear or bilinear lattice models. Our research focuses on the development of an estimation algorithm for a signal process in the lattice models with background additive white noise, and with different assumptions regarding the characteristics of the signal process. We analyze the estimation algorithm and implement it as a stochastic differential equation under the assumption that the Lie algebra, associated with the signal process, can be reduced to a finite dimensional nilpotent algebra. A generalization is given for the case of lattice models, which belong to a class of causal lattices with certain restrictions on input and output signals. The application of adaptive filters for state estimation of the CA3 region of the hippocampus (a common location of the epileptic focus) is discussed. Our areas of application involve two problems: (1) an adaptive estimation of state variables of the hippocampal network, and (2) space identification of the coupled ordinary equation lattice model for the CA3 region.     

Pareto set approximation by the method of adjustable weights and successive lexicographic goal programming

A nonlinear multiobjective optimization problem is considered. Two methods are proposed to generate solutions with an approximately uniform distribution in a Pareto set. The first method is supposed to find the solutions as minimizers of weighted sums of objective functions where the weights are properly selected using a branch and bound type algorithm. The second method is based on lexicographic goal programming. The proposed methods are compared with several metaheuristic methods using two and three-criteria tests and applied problems.     

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

Finding independent sets in a graph using continuous multivariable polynomial formulations

Two continuous formulations of the maximum independent set problem on a graph G=(V,E) are considered. Both cases involve the maximization of an n-variable polynomial over the n-dimensional hypercube, where n is the number of nodes in G. Two (polynomial) objective functions F(x) and H(x) are considered. Given any solution to x₀ in the hypercube, we propose two polynomial-time algorithms based on these formulations, for finding maximal independent sets with cardinality greater than or equal to F(x₀) and H(x₀), respectively. A relation between the two approaches is studied and a more general statement for dominating sets is proved. Results of preliminary computational experiments for some of the DIMACS clique benchmark graphs are presented.     

Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

Equivalent formulations and necessary optimality conditions for the Lennard–Jones problem

The minimization of molecular potential energy functions is one of the most challenging, unsolved nonconvex global optimization problems and plays an important role in the determination of stable states of certain classes of molecular clusters and proteins. In this paper, some equivalent formulations and necessary optimality conditions for the minimization of the Lennard–Jones potential energy function are presented. A new strategy, the code partition algorithm, which is based on a bilevel optimization formulation, is proposed for searching for an extremal Lennard–Jones code. The convergence of the code partition algorithm is proved and some computational results are reported.     

Manganese speciation in human cerebrospinal fluid using CZE coupled to inductively coupled plasma MS

The neurotoxic effects of manganese (Mn) at elevated concentrations are well known. This raises the question, which of the Mn species can cross neural barriers and appear in cerebrospinal fluid (CSF). CSF is the last matrix in a living human organism available for analysis before a compound reaches the brain cells and therefore it is assumed to reflect best the internal exposure of brain tissue to Mn species. A previously developed CE method was modified for separation of albumin, histidine, tyrosine, cystine, fumarate, malate, inorganic Mn, oxalacetate, alpha-keto-glutarate, nicotinamide-dinucleotide (NAD), citrate, adenosine, glutathione, and glutamine. These compounds are supposed in the literature to act as potential Mn carriers. In a first attempt, these compounds were analyzed by CZE-UV to check whether they are present in CSF. The CZE-UV method was simpler than the coupled CZE-inductively coupled plasma (ICP)-dynamic reaction cell (DRC)-MS method and it was therefore chosen to obtain a first overview information. In a second step, the coupled method (CZE-ICP-DRC-MS) was used to analyze, in detail, which of the compounds found in CSF by CZE-UV were actually bound to Mn. Finally, 13 Mn species were monitored in CSF samples, most of them being identified: Mn-histidine, Mn-fumarate, Mn-malate, inorganic Mn, Mn-oxalacetate, Mn-alpha-keto glutarate, Mn-carrying NAD, Mn-citrate and Mn-adenosine. By far the most abundant Mn species was Mn-citrate showing a concentration of 0.7 +/- 0.13 microg Mn/L. Interestingly, several other Mn species can be related to the citric acid cycle.     

Microwave-assisted synthesis of the Schöllkopf chiral auxiliaries: (3S)- and (3R)-3,6-dihydro-2,5-diethoxy-3-isopropyl-pyrazine

A practical and efficient methodology for the laboratory scale preparation of Schöllkopf’s bis-lactim ether chiral auxiliaries (3S)- and (3R)-3,6-dihydro-2,5-diethoxy-3-isopropyl-pyrazine has been developed. The key step is the preparation of the 2,5-diketopiperazine derivative by microwave-assisted heating in water. The protocol avoids reactions at low temperature and the use of high boiling solvents. Only inexpensive and readily available starting materials are required. The bis-lactim ethers were produced in high yields on a multigram scale.     

A diverse human learning optimization algorithm

Human Learning Optimization is a simple but efficient meta-heuristic algorithm in which three learning operators, i.e. the random learning operator, the individual learning operator, and the social learning operator, are developed to efficiently search the optimal solution by imitating the learning mechanisms of human beings. However, HLO assumes that all the individuals possess the same learning ability, which is not true in a real human population as the IQ scores of humans, one of the most important indices of the learning ability of humans, follow Gaussian distribution and increase with the development of society and technology. Inspired by this fact, this paper proposes a Diverse Human Learning Optimization algorithm (DHLO), into which the Gaussian distribution and dynamic adjusting strategy are introduced. By adopting a set of Gaussian distributed parameter values instead of a constant to diversify the learning abilities of DHLO, the robustness of the algorithm is strengthened. In addition, by cooperating with the dynamic updating operation, DHLO can adjust to better parameter values and consequently enhances the global search ability of the algorithm. Finally, DHLO is applied to tackle the CEC05 benchmark functions as well as knapsack problems, and its performance is compared with the standard HLO as well as the other eight meta-heuristics, i.e. the Binary Differential Evolution, Simplified Binary Artificial Fish Swarm Algorithm, Adaptive Binary Harmony Search, Binary Gravitational Search Algorithms, Binary Bat Algorithms, Binary Artificial Bee Colony, Bi-Velocity Discrete Particle Swarm Optimization, and Modified Binary Particle Swarm Optimization. The experimental results show that the presented DHLO outperforms the other algorithms in terms of search accuracy and scalability.