Logoptimization is a new word (obtained by contracting "logic" and "optimization") that stands for "optimization with logic variables". A logic variable is one that can only take two possible values like, e.g., 0 or 1, -1 or 1, 5 or 44, FALSE or TRUE, RED or BLUE, and so on. We assume here for simplicity that these two values are 0 and 1.

A logoptimization problem is the following: a function f (x₁,x₂,x₃,...,x_n ) on n logic variable is given. For each assignment of the values 0 and 1 to the variables x₁,x₂,x₃,...,x_n , the corresponding value of the function f is a real number. Then the problem is to

1. find an assignment for which the value of f is maximal (or minimal) and to
2. provide a mathematical evidence that no other assignment performs better. This evidence is called an optimality certificate.

The number of possible distinct assignments is finite, as there are only 2ⁿ of them. Therefore, finding the solution to the problem is conceptually rather simple, as it amounts to computing the function for each possible pattern of n symbols in the set {0,1} and to taking the pattern that corresponds to the maximal (minimal) value attained by the function f. This simple solution strategy is called complete enumeration or optimization by brute force.

However, since the function 2ⁿ, that gives the number of function evaluations, grows very rapidly as n increases, complete enumeration becomes computationally expensive, or even computationally infeasible, already for moderate values of the parameter n. Unfortunately, the technological advances in the computer industry cannot help much. To make this statement more clear, let us assume that we can get access to an ideal parallel computer with as many computing units as the number of the atoms in the Universe (estimated to be roughly 10¹⁰⁰ ). Let us also assume that each unit is able to perform a billion function evaluations per second. Well, such a colossal technological jewel would take more than 1000 years to solve a logoptimization problem by brute force when the value of the parameter n is only 400!

On the other hands, there are uncountable real problems in Computer Science, Electrical Engineering, Management Science, Decision Science and many other fields, that can be formulated as logoptimization problems with several hundred, or even thousand, variables. Therefore, a great challenge for scientists working in the area of Mathematical Optimization is to devise effective methods that are able solve problems of these sizes in a reasonable amount of computation time.

WE.LOGOPTIMIZE.IT is the web site of a virtual laboratory where a group of European researchers develops methods, algorithms and software for the solution (with optimality certificate) of logoptimization problems. The main purpose of the site is to provide the interested user with a friendly access to a server dedicated to the solution of these problems.

You can browse within the web site to get more insights, references, documentation and other material, and to find out how to use the server to solve your logoptimization problems.

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