SHORTEST PATH SIMPLEX ALGORITHM WITH A MULTIPLE PIVOT RULE: A COMPARATIVE STUDY

This paper introduces a new multiple pivot shortest path simplex method by choosing a subset of non-basic arcs to simultaneously enter into the basis. It is shown that the proposed shortest path simplex method requires O (n) multiple pivots and its running time is O (nm). Results from a computational study comparing the proposed method from previously known methods are reported. The experimental show that the proposed rule is more efficient than the considered shortest path simplex pivot rules.

An Enhanced Memetic Differential Evolution in Filter Design for Defect Detection in Paper Production

This article proposes an Enhanced Memetic Differential Evolution (EMDE) for designing digital filters which aim at detecting defects of the paper produced during an industrial process. Defect detection is handled by means of two Gabor filters and their design is performed by the EMDE. The EMDE is a novel adaptive evolutionary algorithm which combines the powerful explorative features of Differential Evolution with the exploitative features of three local search algorithms employing different pivot rules and neighborhood generating functions. These local search algorithms are the Hooke Jeeves Algorithm, a Stochastic Local Search, and Simulated Annealing. The local search algorithms are adaptively coordinated by means of a control parameter that measures fitness distribution among individuals of the population and a novel probabilistic scheme. Numerical results confirm that Differential Evolution is an efficient evolutionary framework for the image processing problem under investigation and show that the EMDE performs well. As a matter of fact, the application of the EMDE leads to a design of an efficiently tailored filter. A comparison with various popular metaheuristics proves the effectiveness of the EMDE in terms of convergence speed, stagnation prevention, and capability in detecting solutions having high performance.

On pivot rules for simplex method of interior points, and their investigation on Klee-Minty cube

In [25] it was proposed a parametric linear transformation, which is a "convex" combination of the Gauss transformation of elimination method and the Gram-Schmidt transformation of modified orthogonalization process. Using this transformation, in particular, elimination methods were generalized, Dantzig's optimality criterion and simplex method were developed [26]. The essence of the simplex method development is the following. At each sth step the pivot (positive) vector of length Ks is selected, that allows us to move to improved feasible solution after the step of the generalized Gauss-Jordan complete elimination method. In this method the movement to the optimal point takes place over pseudobases, i.e., over interior points. This method is parametric and finite. Since the method is parametric there are various variants to choose the pivot vectors (rules), in the sense of their lengths and indices. In this article we propose three rules, which are the development of Dantzig's first rule. These rules are investigated on the Klee-Minty cube (problem) [14, 31]. It is shown that for two rules the number of steps necessary equals to 2n, and for third rule we obtain the standard simplex method with the largest coefficient rule, i.e., the number of steps for solving this problem is 2n - 1.