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In this chapter, we consider optimality conditions and duality for some multiobjective programming problems with generalized convexity. In particularly, the general multiobjective programming, multiobjective fractional programming and multiobjective variational programming will be discussed.

In many practical situations, we have several possible actions, and we must choose the best action. For example, we must find the best design of an object, or the best control of a plant. The set of possible actions is usually characterized by parameters = (, ..., ), and the result of different actions (controls) is characterized by an ().

We consider the problem of finding the global minimum of a function : → ℝ where ⊆ ℝ is a compact right parallelepiped parallel to the coordinate axes:

The existing image and data compression techniques try to minimize the mean square deviation between the original data () and the compressed-decompressed data (). In many practical situations, reconstruction that only guaranteed mean square error over the data set is unacceptable.

Constrained Optimization Problems (COP’s) are encountered in many scientific fields concerned with industrial applications such as kinematics, chemical process optimization, molecular design, etc. When non-linear relationships among variables are defined by problem constraints resulting in non-convex feasible sets, the problem of identifying feasible solutions may become very hard. Consequently, finding the location of the global optimum in the COP is more difficult as compared to bound-constrained global optimization problems.

This chapter proposes a new interval partitioning method for solving the COP. The proposed approach involves a new subdivision direction selection method as well as an adaptive search tree framework where nodes (boxes defining different variable domains) are explored using a restricted hybrid depth-first and best-first branching strategy. This hybrid approach is also used for activating local search in boxes with the aim of identifying different feasible stationary points. The proposed search tree management approach improves the convergence speed of the interval partitioning method that is also supported by the new parallel subdivision direction selection rule (used in selecting the variables to be partitioned in a given box). This rule targets directly the uncertainty degrees of constraints (with respect to feasibility) and the uncertainty degree of the objective function (with respect to optimality). Reducing these uncertainties as such results in the early and reliable detection of infeasible and sub-optimal boxes, thereby diminishing the number of boxes to be assessed. Consequently, chances of identifying local stationary points during the early stages of the search increase.

The effectiveness of the proposed interval partitioning algorithm is illustrated on several practical application problems and compared with professional commercial local and global solvers. Empirical results show that the presented new approach is as good as available COP solvers.

Interval arithmetic is a valuable tool in numerical analysis and modeling. Interval arithmetic operates with intervals defined by two real numbers and produces intervals containing all possible results of corresponding real operations with real numbers from each interval. An interval function can be constructed replacing the usual arithmetic operations by interval arithmetic operations in the algorithm calculating values of functions. An interval value of a function can be evaluated using the interval function with interval arguments and determines the lower and upper bounds for the function in the region defined by the vector of interval arguments.

Unconstrained pseudo-Boolean optimization is an issue that studied enough now. Algorithms that have been designed and investigated in the area of unconstrained pseudo-Boolean optimization are applied successfully for solving various problems. Particularly, these are local optimization methods [, , ] and stochastic and regular algorithms based on local search for special function classes [, , ]. Moreover, there is a number of algorithms for optimization of functions given in explicit form: Hammer’s basic algorithm that, was introduced in [] and simplified in []; algorithms for optimization of quadratic functions [, , ], etc. Universal optimization methods are also used successfully: genetic algorithms, simulated annealing, tabu search [, ].

In this chapter, Lipschitz univariate constrained global optimization problems where both the objective function and constraints can be multiextremal and non-differentiable are considered. The constrained problem is reduced to a discontinuous unconstrained problem by the index scheme without introducing additional parameters or variables. It is shown that the index approach proposed by R.G. Strongin for solving these problems in the framework of stochastic information algorithms can be successfully extended to geometric algorithms constructing non-differentiable discontinuous minorants for the reduced problem. A new geometric method using adaptive estimates of Lipschitz constants is described and its convergence conditions are established. Numerical experiments including comparison of the new algorithm with methods using penalty approach are presented.

The Hungarian mathematician Farkas Bolyai (1775–1856) published in his principal work (‘Tentamen’, 1832–33 []) a dense regular packing of equal circles in an equilateral triangle (see Fig. 1). He defined an infinite packing series and investigated the limit of (in Latin, the gap in the triangle outside the circles). It is interesting that these packings are not always optimal in spite of the fact that they are based on hexagonal grid packings. Bolyai probably was the first author in the mathematical literature who studied the density of a series of packing circles in a bounded shape.

An idealized thin film when subjected to some constraints acquires length-minimizing properties. The length-minimizing curve of the film may achieve a configuration close to the Steiner minimal tree in the Euclidean plane. The Steiner problem asks for the shortest network that spans a given set of fixed points in the Euclidean plane. The main idea is to use the mathematical model for an idealized wet film, connecting the fixed points with some liquid inside the film. Gradually decreasing the interior area, the film may achieve the globally optimal solution. A system of equations and an algorithm for simulating wet film evolution are presented here. Computational experiments and tests show the abilities of global optimization. The investigation of a simple case illustrates how the film evolution leads up to the global optimum.