By Xiang-Sun Zhang (auth.)

People are dealing with an increasing number of NP-complete or NP-hard difficulties of a combinatorial nature and of a continual nature in monetary, army and administration perform. There are ways that you will improve the potency of attempting to find the ideas of those difficulties. the 1st is to enhance the rate and reminiscence ability of undefined. all of us have witnessed the pc industry's awesome achievements with and software program advancements during the last 20 years. On one hand many desktops, received just a couple of years in the past, are being despatched to trouble-free faculties for kids to profit the ABC's of computing. nevertheless, with monetary, medical and armed forces advancements, it sounds as if the rise of intricacy and the dimensions of newly bobbing up difficulties don't have any finish. all of us notice then that the second one method, to layout reliable algorithms, will certainly make amends for the obstacles in relation to complex difficulties. it's the collective and parallel computation estate of man-made neural internet works that has activated the passion of researchers within the box of machine technological know-how and utilized arithmetic. it truly is tough to claim that man made neural networks are solvers of the above-mentioned limitation, yet no less than they throw a few new gentle at the problems we are facing. We not just expect that there'll be neural pcs with intelligence yet we additionally think that the learn result of synthetic neural networks may result in new algorithms on von Neumann's computers.

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**Extra resources for Neural Networks in Optimization**

**Example text**

It is easy to show that HCP is in NP. Furthermore it is a NP-complete problem. Now consider the following no-yes version of the same problem, or the complement of the HCP: Given a graph G, is G non-Hamilton? This problem is not in NP because if we want to verify a graph being nonHamilton, the only way is to list systematically all circuits and show that none of them covers all of the nodes. This list is a certificate, but has exponential length. The next example is related to the TSP. 29 TSPComplement Given a complete graph G = (V, E), IV I = n with an symmetric distance matrix of nonnegative integers (dij ), and an integer L, find whether for all tours 1r satisfying n 2:::: dj7r(j) > L.

It can be shown ([54]) that if x(t) is stable for t ~ t 0 , it is stable for t ~ t1 > to. 45 is independent of t 0 , the solution is said to be uniformly stable on t ~ t 0 . Any stable solution of an autonomous system must be uniformly stable since the system is invariant with respect to time translation. , the disturbed solution remains a constant distance away. In other words, when x (t) -----+ x*, an equilibrium point, the disturbed solution may converge to another limit point. 26 NEURAL NETWORKS IN OPTIMIZATION When an optimization problem is solved by a neural network which is described by a system of ordinary differential equations, usually the equilibrium points of the system correspond to the set of optimal solutions of the problem.

However the study of the Newton method establishes certain reference plateaus with respect to difficulty of implementation and speed of convergence, by which one can evaluate the efficiencies of other proposed algorithms. 2 GRADIENT METHOD Gradient method is also referred to as steepest descent method. It is one of the oldest and most widely known methods for minimizing a function without constraints. Due to its satisfactory theoretical analysis the method is taken as a standard of reference against which other new algorithms are evaluated.