Decentralized method for computing Pareto solutions in multiparty negotiations

This paper presents a decentralized method for computing Pareto-optimal solutions in multiparty negotiations over continuous issues. The method is based on the well known weighting method which is decomposed by introducing an own decision variable for each decision maker and by applying the dual decomposition method to the resulting problem. The method offers a systematic way for generating some or all Pareto-optimal solutions so that decision makers do not have to know each others' value functions. Under the assumption of quasilinear value function the requirement that a decision maker knows the explicit form for his value function can be relaxed. In that case the decision maker is asked to solve a series of multiobjective programming problems where an additional artificial decision variable is introduced.     

Searching for joint gains in multi-party negotiations

We develop a constructive approach to multi-party negotiations over continuous issues. The method is intended to be used as a mediator's tool for step-by-step creation of joint gains in order to reach a Pareto-optimal agreement. During the mediation process, the parties are only required to answer relatively simple questions concerning their preferences; they do not have to reveal their utility functions completely. The method generates jointly improving directions to move along, and it is a non-trivial generalization of the recently proposed two-party methods. We give a mathematical analysis together with a numerical example, but also a practical basis for negotiation support in real-world settings.     

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.     

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 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.     

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 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.     

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.     

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.     

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 constrained robust proposal for mixture modeling avoiding spurious solutions

The high prevalence of spurious solutions and the disturbing effect of outlying observations in mixture modeling are well known problems that pose serious difficulties for non-expert practitioners of this kind of models in different applied areas. An approach which combines the use of Trimmed Maximum Likelihood ideas and the imposition of restrictions on the maximization problem will be presented and studied in this paper. The proposed methodology is shown to have nice mathematical properties as well as good performance in avoiding the appearance of spurious solutions in a quite automatic manner.     

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.     

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.     

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

A method is presented for generating a well-distributed Pareto set in nonlinear multiobjective optimization. The approach shares conceptual similarity with the Physical Programming-based method, the Normal-Boundary Intersection and the Normal Constraint methods, in its systematic approach investigating the objective space in order to obtain a well-distributed Pareto set. The proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space. It is shown that the proposed modification allows the method to generate an even representation of the entire Pareto surface. The generation is performed for both convex and nonconvex Pareto frontiers. A simple algorithm has been proposed to remove local Pareto solutions. The suggested approach has been verified by several test cases, including the generation of both convex and concave Pareto frontiers.     

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.