Games with multiple payoffs

The traditional theories of decision making and games are based on an assumption that prevents their broader practical utilization:a single dimensional payoff. In reality, any alternative is likely to imply more than one payoff, e.g. not only costs but also time, price, quality, safety, maintainability, productivity, etc. Similarly, the Theory of Games faces difficulties as we attempt to apply it to the conflict situations of the social and business environment. The assumption that a single dimensional payoff, like money, points or chips, is a realistic consequence of individual moves or strategies, is difficult to sustain. In this short paper, concepts of nondominated solutions and a decomposition of parametric spaces are used to formulate and resolvedecision problems with vector payoffs. Both Games against Nature and Two-Person, Zero-Sum frameworks are considered. A concept of compromise solutions is introduced to help the decision maker with further reduction of the nondominated set. Some numerical examples are given.     

On duality principles for scalar and vectorial multi-well variational problems

This work develops duality applicable to vectorial problems in the calculus of variations. The results can be used to compute solutions related to the relaxation of non-convex problems, such as vectorial phase transition models, particularly as the primal approaches have no solutions in the classical sense. In this latter case, the solution of the dual problem is a weak cluster point of minimizing sequences for the original formulation. Here we are not restricted to two or three wells only, the results developed are suitable for more general situations.     

The General Problem of <Emphasis Type="Italic">m</Emphasis> Persons and <Emphasis Type="Italic">n</Emphasis> Positions for Short- and Long-Term Decision-Making

This paper discusses the well-known problems of m candidates and n positions usually called term assignment problems. The paper considers these problems when decisions for short and long term policies must be taken and also in extreme situations, such as emergency cases. Assignment and Knapsack problems, Heirarchy Analysis, Graphs Theory and Metric Spaces using Dynamic Programming are used on a computer.     

Problems and results in extremal combinatorics—II

Extremal Combinatorics is one of the central areas in Discrete Mathematics. It deals with problems that are often motivated by questions arising in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics considered here include questions in Extremal Graph Theory, Polyhedral Combinatorics and Probabilistic Combinatorics. This is not meant to be a comprehensive survey of the area, it is merely a collection of various extremal problems, which are hopefully interesting. The choice of the problems is inevitably biased, and as the title of the paper suggests, it is a sequel to a previous paper [N. Alon, Problems and results in extremal combinatorics—I, Discrete Math. 273 (2003), 31-53.] of the same flavor, and hopefully a predecessor of another related future paper. Each section of this paper is essentially self contained, and can be read separately.     

The Regularity for Solutions of Variational Inequalities

The regularity for solutions of elliptic equations is rather perfectly solved. But it does not so perfect for that of elliptic variational inequalities. In literature only different special situations are considered. Now the boundedness, C^{0,λ} continuity and C^{1,α} regularity are proved for solutions of one-sided obstacle problems under more general structural conditions, in which the growth orders of u are permitted to reach the critical exponents and the growth order ϒ of the gradient in D is permitted to be super critical as 1 < p < n.     

A Bliss-Type Multiplier Rule for Constrained Variational Problems with Time Delay

This paper gives a Bliss-type multiplier rule for general constrained variational problems with time delay. Our formulation allows the time delay terms and their derivatives to be present in both the objective functional and the equality and inequality constraint equations. Our major results include a Lagrange Multiplier Rule involving Euler–Lagrange equations for delay differential equations and transversality conditions. Of special note is that these results will be used in later work by the authors to obtain new analytical techniques to solve general constrained problems involving delay differential equations for the Calculus of Variations and for Optimal Control Theory. They will also lead to new general, accurate, and efficient numerical methods to solve these important problems where no such methods exist at this time. Thus, for the first time, we will be able to solve this important class of problems obtaining analytic solutions for simpler problems and accurate numerical results for more complex problems when such a solution exists.     

A method to bypass the lack of solutions in MinSup problems under quasi-equilibrium constraints

MinSup problems with constraints described by quasi-equilibrium problems are considered in Banach spaces. The solutions set of such problems may be empty even in very good situations, so the aim of this paper is twofold. First, we determine appropriate regularizations (called inner regularizations) which allow to reach the value of the original problem. Then, among these regularizations we identify those which allow to bypass the lack of exact solutions to these problems by a suitable concept of “viscosity” solution whose existence is then proved under reasonable assumptions.     

The new challenges of management in a wired world

The old causal models used in decision support systems are not adequate anymore to represent the evolution of systems presenting the high rhythm of interactions of modern organisations. Emergent properites or even chaotic situations are resulting from the non linear dynamic of these systems. The “Theory of Complexity” of the Santa Fe Institute offers new approaches to these problems.     

On the existence of solutions to generalized quasi-equilibrium problems

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65–76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.     

Existence of Solutions of Bifunction-Set Optimization Problems in Metric Spaces

Journal of Optimization Theory and Applications - Sufficient conditions are given for the existence of solutions of bifunction-set optimization problems, whose underlying sets are complete metric...     

On a System of Differential Equations Connected with the Gravitational Instability of a Multicomponent Medium in Newtonian Cosmology

The gravitational clustering of a multicomponent medium in an expanding universe has been considered by many authors in recent years. The system of differential equations associated with such a multicomponent medium is generalized in this article by incorporating arbitrary parameters. Explicit analytic solutions of this generalized system are given for various situations. Physical interpretations of the solutions are not considered, but the various solutions of physical problems obtained by various authors can be seen to be special cases of the general solutions given in this article.     

A Local Boundedness Result for a Class of Obstacle Problems with Non-Standard Growth Conditions

Journal of Optimization Theory and Applications - We prove the local boundedness for solutions to a class of obstacle problems with non-standard growth conditions. The novelty here is that we are...     

Characterizing the efficient points without closedness or free-disposability

Many multicriteria problems in economics and finance require that efficient solutions be found. A recent contribution to production theory established a characterization of efficient points under closedness and free-disposability (Bonnisseau and Crettez in Econ Theory 31(2):213–223, 2007, Theorem 1). However, as will be shown using a number of examples, these results cannot be applied to simple and plausible production sets, nor can they be extended to other classic multicriteria problems such as those arising in optimal portfolio theory and bargaining theory. To address these limitations, a reformulation of the above theorem without closedness or free-disposability is proposed. This enables efficient solutions for a wider range of multicriteria problems to be identified.     

Didactical contract and responsiveness to didactical contract: a theoretical framework for enquiry into students’ creativity in mathematics

One of the manifestations of learning is the student’s ability to come up with original solutions to new problems. This ability is one of the criteria by which the teacher may assess whether the student has grasped the taught mathematics. Obviously, a teacher can never teach the ability to invent new solutions (at least not directly): he/she can ask for it, expect it, encourage it, but cannot require it. This is one of the fundamental paradoxes of the whole didactical relationship, which Guy Brousseau modelled in one of the best-known concepts in didactics of mathematics: the didactical contract. The teacher cannot be confident that the student will learn exactly what the teacher intended to teach and hence the student must re-create it on the basis of what he/she already knows. In this paper, the importance and the role of situations affording mathematical creativity (in the sense of production of original solutions to unusual situations) are demonstrated. The authors present an experiment (with 9–10-year-old children) that makes it possible to show how certain situations are more favourable (for all children) to express some characteristics of mathematical creativity.     

Existence and Uniqueness for Some 3rd Order Dissipative Problems with Various Boundary Conditions

We briefly present new results on the existence and uniqueness of solutions u of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in Superconductor Theory and in the Theory of Viscoelastic Materials.     

Metrically well-set minimization problems

A concept of well-posedness, or more exactly of stability in a metric sense, is introduced for minimization problems on metric spaces generalizing the notion due to Tykhonov to situations in which there is no uniqueness of solutions. It is compared with other concepts, in particular to a variant of the notion after Hadamard reformulated via a metric semicontinuity approach. Concrete criteria of well-posedness are presented, e.g., for convex minimization problems.     

Case Studies for a First-Order Robust Nonlinear Programming Formulation

In this paper, we conduct three case studies to assess the effectiveness of a recently proposed first-order method for robust nonlinear programming [Zhang, Y.: J. Optim. Theory Appl. 132, 111–124 (2007)]. Three robust nonlinear programming problems were chosen from the literature using the criteria that results calculated using other methods must be available and the problems should be realistic, but fairly simple. Our studies show that the first-order method produced reasonable solutions when the level of uncertainty was small to moderate. In addition, we demonstrate a method for leveraging a theoretical result to eliminate constraint violations. Since the first-order method is relatively inexpensive in comparison to other robust optimization techniques, our studies indicate that, under moderate uncertainty, the first-order approach may be more suitable than other methods for large problems. The authors recognize funding from NSF Grants DMS-0405831 and DMS-0240058.     

Some properties of slowly decaying oscillatory solutions of KP

In this paper a class of solutions of KP is discussed. Theory of these solutions, referred to here as breathers, is developed and it is shown that a subclass of these solutions can be used to construct more general solutions of KP similarly to how the functions e^i(λx+νy) are used to perform the same task in the theory of Fourier transform. Nonlinear superposition formula for general solutions of KP similar to the Fourier expansion formula is considered.     

Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications

Journal of Optimization Theory and Applications - In this paper, we give some properties concerned with weak minimal solutions of nonconvex set optimization problems. We also give some properties...     

The Solution of Massive Generalized Set Partitioning Problems in Aircrew Rostering

The scheduling and rostering of personnel is a problem that occurs in many organizations. Aircrew scheduling has attracted considerable attention with many heuristic methods being proposed, but in recent times set partitioning optimization methods have become more popular. The aircrew rostering problem is discussed and formulated as a generalized set partitioning model. Because of the extremely large optimization models that are generated in practical situations, some special computational techniques have been developed to produce solutions efficiently. These techniques are used to solve problems arising from an airline application in which set partitioning models with more than 650 constraints and 200 000 binary variables are generated. The solutions are produced on a Motorola 68020 microprocessor in little more than three hours.