Constraint proposal method for computing Pareto solutions in multi-party negotiations

The constraint proposal method for computing Pareto-optimal solutions is extended to multi-party negotiations. In the method a neutral coordinator assists decision makers in finding Pareto-optimal solutions so that the elicitation of the decision makers' value functions is not required. During the procedure the decision makers have to indicate their most preferred points on different sets of linear constraints. The method can be used to generate either one Pareto-optimal solution dominating the status quo solution of the negotiation or an approximation to the Pareto frontier. In the latter case a distributive negotiation among the efficient agreements can be carried out afterwards.     

On homotopy continuation method for computing multiple solutions to the Henon equation

Motivated by numerical examples in solving semilinear elliptic PDEs for multiple solutions, some properties of Newton homotopy continuation method, such as its continuation on symmetries, the Morse index, and certain functional structures, are established. Those results provide useful information on selecting initial points for the method to find desired solutions. As an application, a bifurcation diagram, showing the symmetry/peak breaking phenomena of the Henon equation, is constructed. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A power series method for computing singular solutions to nonlinear analytic systems

Summary Given a system of analytic equations having a singular solution, we show how to develop a power series representation for the solution. This series is computable, and when the multiplicity of the solution is small, highly accurate estimates of the solution can be generated for a moderate computational cost. In this paper, a theorem is proven (using results from several complex variables) which establishes the basis for the approach. Then a specific numerical method is developed, and data from numerical experiments are given.     

Sufficient conditions for global weak Pareto solutions in multiobjective optimization

In this paper we derive new sufficient conditions for global weak Pareto solutions to set-valued optimization problems with general geometric constraints of the type $$\begin{aligned} \text{ maximize}\quad F(x) \quad \text{ subject} \text{ to}\quad x\in \Omega , \end{aligned}$$ where $F: X\rightrightarrows Z$ is a set-valued mapping between Banach spaces with a partial order on $Z$ . Our main results are established by using advanced tools of variational analysis and generalized differentiation; in particular, the extremal principle and full generalized differential calculus for the subdifferential/coderivative constructions involved. Various consequences and refined versions are also considered for special classes of problems in vector optimization including those with Lipschitzian data, with convex data, with finitely many objectives, and with no constraints.     

A descent method for Pareto optimization

A descent method on a closed set X of a Hilbert space, adapted to the multi-objective optimization, is presented. After solving a differential inclusion, the limit points of the solutions are used to characterize a critical set, which contains the set of Pareto optima. Under suitable assumptions existence of a Pareto optimum is proved. Then the Lusternik-Schnirelman theory is generalized to this framework and the critical set is related to the topological properties of X.     

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

We consider fair allocation of indivisible items under additive utilities. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. The result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stronger concepts.     

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.     

Distributed computation of Pareto solutions inn-player games

The problem of computing Pareto optimal solutions with distributed algorithms is considered inn-player games. We shall first formulate a new geometric problem for finding Pareto solutions. It involves solving joint tangents for the players' objective functions. This problem can then be solved with distributed iterative methods, and two such methods are presented. The principal results are related to the analysis of the geometric problem. We give conditions under which its solutions are Pareto optimal, characterize the solutions, and prove an existence theorem. There are two important reasons for the interest in distributed algorithms. First, they can carry computational advantages over centralized schemes. Second, they can be used in situations where the players do not know each others' objective functions.     

Advanced computing solutions for health care and medicine

This guest editorial introduces the special issue on “Advanced Computing Solutions for Health Care and Medicine”. The goal of this special issue was to collect high quality papers describing the application of computer science methods and techniques to main health care and clinical problems, resulting in high performance applications or prototypes for medical and clinical environments. The special issue touched different health informatics hot topics and is organized in four sections: (i) clinical decision support systems; (ii) biomedical imaging; (iii) high performance computing and biomedical simulations; (iv) bioinformatics data analysis.     

Probabilistic clustering via Pareto solutions and significance tests

The present paper proposes a new strategy for probabilistic (often called model-based) clustering. It is well known that local maxima of mixture likelihoods can be used to partition an underlying data set. However, local maxima are rarely unique. Therefore, it remains to select the reasonable solutions, and in particular the desired one. Credible partitions are usually recognized by separation (and cohesion) of their clusters. We use here the p values provided by the classical tests of Wilks, Hotelling, and Behrens–Fisher to single out those solutions that are well separated by location. It has been shown that reasonable solutions to a clustering problem are related to Pareto points in a plot of scale balance vs. model fit of all local maxima. We briefly review this theory and propose as solutions all well-fitting Pareto points in the set of local maxima separated by location in the above sense. We also design a new iterative, parameter-free cutting plane algorithm for the multivariate Behrens–Fisher problem.     

Improving a method for computing non-dominant solutions of certain second-order recurrence relations of Poincaré-type

Summary In this paper we present a method of convergence acceleration for the calculation of non-dominant solutions of second-order linear recurrence relations for which the coefficients satisfy certain asymptotic conditions. It represents an improvement of the method recently proposed by Jacobsen and Waadeland [3, 4] for limit periodic continued fractions. For continued fractions the method corresponds to a repeated application of the Bauer-Muir transformation. Some examples and a generalization to non-homogeneous recurrence relations are given.     

A recursive method for computing interpolants

In this paper, we describe a recursive method for computing interpolants defined in a space spanned by a finite number of continuous functions in R^d${\mathbb{R}}^d$. We apply this method to construct several interpolants such as spline interpolants, tensor product interpolants and multivariate polynomial interpolants. We also give a simple algorithm for solving a multivariate polynomial interpolation problem and constructing the minimal interpolation space for a given finite set of interpolation points.     

A new method for solving Pareto eigenvalue complementarity problems

In this paper, we introduce a new method, called the Lattice Projection Method (LPM), for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems. The LPM is compared to SNM_min and SNM_FB, two methods widely discussed in the literature for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool (Dolan, Moré in Math. Program. 91:201–213, 2002). The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market (Boisvert et al. in The quality of numerical software: assessment and enhancement, pp. 125–137, 1997), are computing time, number of iterations, number of failures and maximum number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.     

Hybrid schemes with high-order multioperators for computing discontinuous solutions

Results are presented concerning high-order multioperator schemes and their monotonized versions as applied to the computation of discontinuous solutions. Two types of hybrid schemes are considered. Solutions of several test problems, including those with extremely strong discontinuities, are presented. An example of solving the Navier-Stokes equations at low supersonic Mach numbers by applying multioperator schemes without monotonization is given.     

Stability analysis of the Pareto optimal solutions for some vector boolean optimization problem

In this article we consider the boolean optimization problem of finding the set of Pareto optimal solutions. The vector objectives are the positive cuts of linear functions to the non-negative semi-axis. Initial data are subject to perturbations, measured by the l ₁-norm in the parameter space of the problem. We present the formula expressing the extreme level (stability radius) of such perturbations, for which a particular solution remains Pareto optimal.     

A new Pareto set generating method for multi-criteria optimization problems

This paper proposes a new classical method to capture the complete Pareto set of a multi-criteria optimization problem (MOP) even without having any prior information about the location of Pareto surface. The solutions obtained through the proposed method are globally Pareto optimal. Moreover, each and every global Pareto optimal point is within the attainable range. This paper also suggests a procedure to ensure the proper Pareto optimality of the outcomes if slight modifications are allowed in the constraint set of the MOP under consideration. Among the set of all outcomes, the proposed method can effectively detect the regions of unbounded trade-offs between the criteria, if they exist.     

An effective method for computing regression quantiles

Regression quantiles were introduced in Koenker & Bassett[7] as quantities of interest in developing robust estimationprocedures. They can be computed by linear programming combinedwith post optimality techniques. Here an effective alternativeis presented based on the reduced gradient algorithm for l₁fitting as described in [8] combined with a piecewise linearhomotopy. There is a close connection between the two approaches(the new method essentially describes what is going on in thelinear programming), but it is argued that the new approachis preferable. Its robustness as a computational procedure isillustrated by several examples which give rise to a varietyof different behaviours.