7 J an 2 00 4 On the Global Minimization of the Value - at - Risk ∗ Jong - Shi Pang

In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined bounding procedures can be used to compute the latter value to global optimality. A numerical example is provided to illustrate the methodology. Dedication. With great pleasure we dedicate this paper to a respected pioneer of our field, Professor Olvi L. Mangasarian, on the occasion of his 70th birthday. The two topics of this paper, LPECs and smoothing methods, are examples of the vast contributions that Olvi has made in optimization, which have benefited us in many ways and which will continue to benefit us in the future. Happy 70th birthday, Olvi!