Improved ε-Constraint Method for?Multiobjective Programming

In this paper, we revisit one of the most important scalarization techniques used in multiobjective programming, the ε-constraint method. We summarize the method and point out some weaknesses, namely the lack of easy-to-check conditions for properly efficient solutions and the inflexibility of the constraints. We present two modifications that address these weaknesses by first including slack variables in the formulation and second elasticizing the constraints and including surplus variables. We prove results on (weakly, properly) efficient solutions. The improved ε-constraint method that we propose combines both modifications. The research of M. Ehrgott was partially supported by University of Auckland Grant 3602178/9275 and by Deutsche Forschungsgemeinschaft Grant Ka 477/27-1. The research of S. Ruzika was partially supported by Deutsche Forschungsgemeinschaft Grant HA 1795/7-2. The authors thank the anonymous referees, whose comments helped improving the presentation of the paper including a shorter proof of Theorem 3.1.     

Convergence of stochastic search algorithms to finite size pareto set approximations

In this work we investigate the convergence of stochastic search algorithms toward the Pareto set of continuous multi-objective optimization problems. The focus is on obtaining a finite approximation that should capture the entire solution set in a suitable sense, which will be defined using the concept of ε-dominance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We propose and analyse two different archiving strategies which lead to a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting Pareto set approximation.      

ProperD-solutions of multiobjective programming problems with set functions

A special class of solutions for multiobjective programming problems with set functions is considered. A subset of nondominated solutions, called properD-solution set, with respect to a given domination structure is characterized under two situations, with and without inequality constraints.The authors greatly appreciate valuable comments received from the referees.     

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial order is the intersection of its total (linear) refinements or extensions. Since in mathematical social sciences or, more general, in any theory that combines the concepts of topology and order one is mainly interested in continuous total orders or preorders in this paper some aspects of a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed. In particular, necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Dushnik-Miller theorem will be presented. In addition, it will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general. Further, necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Szpilrajn theorem will be presented.      

Properties of the solution set of nonlinear evolution inclusions

In this paper we examine nonlinear, nonautonomous evolution inclusions defined on a Gelfand triple of spaces. First we show that the problem with a convex-valued,h*-usc inx orientor fieldF(t, x) has a solution set which is anR _δ-set inC(T, H). Then for the problem with a nonconvex-valuedF(t, x) which ish-Lipschitz inx, we show that the solution set is path-connected inC(T, H). Subsequently we prove a strong invariance result and a continuity result for the solution multifunction. Combining these two results we establish the existence of periodic solutions. Some examples of parabolic partial differential equations with multivalued terms are also included. This work was done while the authors were visiting the Florida Institute of Technology.     

An approach to the Gummel map by vector extrapolation methods

The numerical approximation of nonlinear partial differential equations requires the computation of large nonlinear systems, that are typically solved by iterative schemes. At each step of the iterative process, a large and sparse linear system has to be solved, and the amount of time elapsed per step grows with the dimensions of the problem. As a consequence, the convergence rate may become very slow, requiring massive cpu-time to compute the solution. In all such cases, it is important to improve the rate of convergence of the iterative scheme. This can be achieved, for instance, by vector extrapolation methods. In this work, we apply some vector extrapolation methods to the electronic device simulation to improve the rate of convergence of the family of Gummel decoupling algorithms. Furthermore, a different approach to the topological ε-algorithm is proposed and preliminary results are presented.     

Acceptable optimality in linear fractional programming with fuzzy coefficients

Based on the specified grades of satisfaction, we propose two new concepts of (α, β)-acceptable optimal solution and (α, β)-acceptable optimal value of a fuzzy linear fractional programming problem with fuzzy coefficients, and develop a method to compute them. An example is provided to demonstrate the method.     

A note on the generalized min-sum set cover problem

In this paper, we consider the generalized min-sum set cover problem, introduced by Azar, Gamzu, and Yin (STOC 2009). Bansal, Gupta, and Krishnaswamy (SODA 2010) give a 485-approximation algorithm for the problem. We are able to alter their algorithm and analysis to obtain a 28-approximation algorithm, improving the performance guarantee by an order of magnitude. We use concepts from α-point scheduling to obtain our improvements.     

Noncommutative versions of the Singer-Wermer conjecture with linear left ϑ-derivations

The noncommutative Singer-Wermer conjecture states that every linear （possibly unbounded） derivation on a （possibly noncommutative） Banach algebra maps into its Jacobson radical. This conjecture is still an open question for more than thirty years. In this paper we approach this question via linear left θ-derivations.     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.      

Convergence of a sequence of weakly regular set functions

The present paper is devoted to generalizations of the Dieudonné theorem claiming that the convergence of sequences of regular Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all Borelian subsets of this space. The Dieudonné theorem is proved in the case for which the set functions are weakly regular, nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary σ-topological space, and take values in a uniform space. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 103–110, July, 1997. Translated by O. V. Sipacheva     

Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning

In this paper, we propose a modification of Benson’s algorithm for solving multiobjective linear programmes in objective space in order to approximate the true nondominated set. We first summarize Benson’s original algorithm and propose some small changes to improve computational performance. We then introduce our approximation version of the algorithm, which computes an inner and an outer approximation of the nondominated set. We prove that the inner approximation provides a set of -nondominated points. This work is motivated by an application, the beam intensity optimization problem of radiotherapy treatment planning. This problem can be formulated as a multiobjective linear programme with three objectives. The constraint matrix of the problem relies on the calculation of dose deposited in tissue. Since this calculation is always imprecise solving the MOLP exactly is not necessary in practice. With our algorithm we solve the problem approximately within a specified accuracy in objective space. We present results on four clinical cancer cases that clearly illustrate the advantages of our method.     

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

In this paper we consider the Lane–Emden problem adapted for the p-Laplacian

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .      

Lower Semicontinuity and Upper Semicontinuity of the Solution Sets and Approximate Solution Sets of Parametric Multivalued Quasivariational Inequalities

We consider the semicontinuity of the solution set and the approximate solution set of parametric multivalued quasivariational inequalities in topological vector spaces. Three kinds of problems arising from the multivalued situation are investigated. A rather complete picture, which is symmetric for the two kinds of semicontinuity (lower and upper semicontinuity) and for the three kinds of multivalued quasivariational inequality problems, is supplied. Moreover, we use a simple technique to prove the results. The results obtained improve several known ones in the literature. This research was partially supported by the National Basic Research Program in Natural Sciences of Vietnam. The final part of this work was completed during a stay of the first author at the Department of Mathematics, University of Pau, Pau, France, and its hospitality is acknowledged.     

Practical Study on the Fuzzy Risk of Flood Disasters

The simplest way to perform a fuzzy risk assessment is to calculate the fuzzy expected value and convert fuzzy risk into non-fuzzy risk, i.e., a crisp value. In doing so, there is a transition from the fuzzy set to the crisp set. Therefore, the first step is to define an α level value, and then select the elements x with a subordinate degree A(x)≥α. The higher the value of α, the lower the degree of uncertainty—the probability is closer to its true value. The lower the value of α, the higher the degree of uncertainty—this results in a lower probability serviceability. The possibility level α is dependant on technical conditions and knowledge. A fuzzy expected value of the possibility-probability distribution is a set with and as its boundaries. The fuzzy expected values and of a possibility-probability distribution represent the fuzzy risk values being calculated. Therefore, we can obtain a conservative risk value, a venture risk value and a maximum probability risk value. Under such an α level, three risk values can be calculated. As α adopts all values throughout the set [0,1], it is possible to obtain a series of risk values. Therefore, the fuzzy risk may be a multi-valued risk or set-valued risk. Calculation of the fuzzy expected value of flood risk in the Jinhua River basin has been performed based on the interior-outer set model. Selection of an α value depends on the confidence in different groups of people, while selection of a conservative risk value or venture risk value depends on the risk preference of these people.     

Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods

We study constrained and unconstrained optimization programs for nonconvex maximum eigenvalue functions. We show how second order techniques may be introduced as soon as it is possible to reliably guess the multiplicity of the maximum eigenvalue at a limit point. We examine in which way standard and projected Newton steps may be combined with a nonsmooth first-order method to obtain a globally convergent algorithm with a fair chance to local superlinear or quadratic convergence. Dedicated to R. T. Rockafellar on the occasion of his 70th anniversary