In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input-output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used...

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Introduction Complexity theory refers to the asymptotic analysis of problems and algorithms. How efficient is an algorithm for a particular optimization problem, as the number of variables gets large? Are there problems for which no efficient algorithm exists? These are the questions that complexity theory attempts to address. The theory originated in work by Hartmanis and Stearns (1965). By now there is much known about complexity issues in nonlinear optimization. In particular, our recent book Vavasis (1991) contains all the details on many of the results surveyed in this chapter. We begin the discussion with a look at convex problems in the next section. These problems generally have efficient algorithms. In Section 3 we study the complexity of two nonconvex problems that also have efficient algorithms because of special structure. In Section 4, we look into hardness results (proofs of the nonexistence of efficient algorithms) for general nonconvex problems. Finally, in Sec

We present an interior point approach to the zero-one integer programming feasibility problem based on the minimization of an appropriate potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such that each consecutive point reduces the value of the potential function. An integer solution (not necessarily feasible) is generated at each iteration by a rounding scheme. The direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid. We illustrate the approach by considering a class of difficult NP-complete problems: finding a maximum independent set of a dense random graph. Some implementation details are discussed and preliminary computational results are presented. We solve several large independent set problems in graphs having up to 1000 vertices and over 250,000 edges. Key words: Integer programming, interior point method, maxim...