Valid inequalities for sets defined by multilinear functions

Transcription

1 Valid inequalities for sets defined by multilinear functions Pietro Belotti 1 Andrew Miller 2 Mahdi Namazifar Lehigh University 200 W Packer Ave Bethlehem PA 18018, USA belotti@lehighedu 2 Institut de Mathématiques de Bordeaux (IMB 351 cours de la Libération Talence, France AndrewMiller@mathu-bordeaux1fr Abstract Mechanical Engineering Building 1513 University Avenue Madison WI 53706, USA namazifar@wiscedu We describe a class of valid inequalities for the bounded, nonconvex set described by all points within a (n + 1-dimensional hypercube whose (n + 1-st coordinate is equal to the product of the first n coordinates, for any n 2 This set can be defined as M n = {x R n+1 : x n+1 = x 1 x n,l i x i u i i =1, 2,n +1}, with l i and u i constants Approximating the convex hull of M n through linear inequalities is essential to a class of exact solvers for nonconvex optimization problems, namely those which use Linear Programming relaxations to compute a lower bound on the problem Together with the well-known McCormick inequalities, these inequalities are valid for the convex hull of M 2 There are infinitely many such inequalities, given that the convex hull of M 2 is not, in general, a polyhedron The generalization to M n for n>2 is straightforward, and allows us to define strengthened relaxations for these higher dimensional sets as well Keywords: nonconvex optimization, multilinear functions, convex hull 13

2 14 Belotti, Miller, Namazifar 1 Multilinear sets Consider the following nonconvex, bounded set n M n = {x R n+1 : x n+1 = x i, l i x i u i,i=1, 2,n+1}, i=1 where l and u are (n + 1-vectors We assume 0 < l i <u i for all i =1, 2,n+1 The set M n is nonconvex as the function ξ n (x = n i=1 x i is neither convex nor concave its epigraph epi(m ={x R n+1 : x n+1 n i=1 x i, 0 < l i x i u i,i= 1, 2,n} and its hypograph hyp(m ={x R n+1 : x n+1 n i=1 x i, 0 < l i x i u i,i=1, 2,n} are nonconvex sets Assuming M n is contained in the first orthant, trivial bounds for x n+1 are given by n i=1 l i and n i=1 u i, respectively, and denoted by l n+1 and ū n+1 In general, l n+1 l n+1 <u n+1 ū n+1 ; in the remainder, we use the notation Mn for the set M n where l n+1 = l n+1 and u n+1 =ū n+1 We are interested in developing a convex superset C n of M n defined by a system of linear inequalities, therefore we seek a polyhedral set C n M n Our interest is motivated by the nonconvex Optimization problem P :min{cx : x X}, where X is, in general, a nonconvex set In order to solve problems like P to optimality, one needs to implicitly enumerate all local optima, for instance by branchand-bound algorithms The bounding algorithms in such approaches often relies on a linear relaxation of the nonconvex problem [3, 4], thus benefiting from tighter linear relaxations 2 Linear inequalities for n =2 Here we develop convex inequalities for the set M 2 = {(x 1,,x 3 [l, u] :x 3 = x 1 } Although this is the simplest case, many considerations generalize easily to the case when n>2 21 Unbounded x 3 The following linear relaxation of M 2 was introduced by McCormick [2] and shown to be its convex hull by Al-Khayyal and Falk [1]: x 3 x 1 + x 3 x 1 + x 3 x 1 + x 3 x 1 +, and is depicted in Figure 1a for convenience The shaded tetrahedron is the (polyhedral convex hull, while M 2 is shown in colors 22 Nontrivial lower and upper bound for x 3 We consider first a finite lower bound, l 3 > l 3 = The darker area in Figure 1b shows the projection of M 2 onto (x 1,, that is, the set P 2 = {(x 1, R 2 : l i x i u i,i =1, 2,x 1 l 3 } It is safe to assume here that l 3 and l 3, as otherwise a valid lower bound for x 1 (resp would be l 3 / > (resp l 3 / >, or equivalently, the upper left (resp the lower right corner of the

3 Inequalities for multilinear sets 15 x 3 x f x x 1 (a Convex hull of M 2 0 x 1 (b Projections of M 2 with non-trivial lower bound l 3 Figure 1 bounding box in Figure 1b would be cut out by the convex set x 1 l 3 Similarly we assume that u 3 and u 3 Projecting M 2 onto (x 1, gives a convex set P 2 Consider a point x on the curve x 1 = l 3 with x 1 and x 2 = l 3 /x 1 The tangent to the curve x 1 = l 3 at x gives a linear inequality a 1 (x 1 x 1+a 2 ( x 2 0, that is valid for P 2 The coefficients a 1 and a 2 are given by the gradient of the function ξ 2 (x =x 1 at x, ie, a 1 = π x 1 x = x 2 and a 2 = π x = x 1 The lighter area within the bounding box above the tangent line is the set of points satisfying the above linear constraint, which we rewrite here: (31 x 2(x 1 x 1+x 1( x 2 0 As this inequality is valid within P 2 and is independent from x 3,itisalsovalid for M 2 We can lift it as follows: the inequality (32 x 2(x 1 x 1+x 1( x 2+a(x 3 l 3 0 is clearly valid for x 3 = l 3 ForittobevalidforM 2, g(a 3 =min{x 2(x 1 x 1+x 1( x 2+a(x 3 l 3 :(x 1,,x 3 M 2 } 0 must hold It is easy to show that g(a 3 =0ifa 0 (a global optimum is given by (x 1,x 2, hence a<0 in all lifted inequalities (32 We next show how to calculate a First we will show how to get this coefficient and then we will prove that the inequalities we get are not dominated by any other valid inequality To find the coefficient a in the inequality (32, intuitively we know that the plane (33 x 2(x 1 x 1+x 1( x 2+a(x 3 l 3 =0 should touch the curve x 1 = u 3 at exactly one point and first we want to find this point Let s call this point ( x 1, If for a moment we disregard the bounds on x 1 and, the fact that the plane (33 touches the curve ( x 1, at exactly one point means that the plane would be tangent to the curve x 1 = u 3 at that point This means that the gradient of the curve at ( x 1, is parallel to the projection of the normal to the plane onto the (x 1, space The gradient of the curve x 1 = u 3 at

4 16 Belotti, Miller, Namazifar ( x 1, is(, x 1 and the projection of the normal to the plane onto the (x 1, space is (x 2,x 1 As a result we will have: (34 R : = x 2, x 1 = x 1 But we know that x 1 = u 3, so we will have x 1 = u 3 = 2 x 1x 2 = 2 l 3, and as a result u3 (35 = l 3 On the other hand we know that ( x 1, is on the plane (33, which implies As a result, x 2( x 1 x 1+x 1( x 2+a(u 3 l 3 =0, (36 x 2(x 1 x 1+x 1(x 2 x 2+a(u 3 l 3 =0 By (35 and (36, the value for a is 2(1 l l 3 3 u 3 l 3 Notice that the value of a does not depend on the value of x 1 and x 2 and is less than zero Intuitively this value for a will give a tight valid inequality for M 2 if x 1 and (the proof will come later But if ( x 1, is not in the domain of (x 1, we need to find a point on the curve x 1 = u 3 at which the plane (33 touches the curve Intuitively this point would be one of the end points of the curve x 1 = u 3 ; x 1, In fact that point will be the one which is closer to ( x 1, Again consider the curve x 1 = u 3 On this curve we know that: q u3 x 1 u 3 = ; u 3 = ; x 1 u 3 =ū 1 ; u 3 =ū 2 Then the real bounds over x 1 and on the curve x 1 = u 3 would be l 1 x 1 u 1, l 1 =max(,, and u 1 =min(ū 1, ; l 2 u 2, l 2 =max(,, and u 2 =min(ū 2, However, based on the assumptions on the upper bound of x 3 (see Section 22 (37 l 1 = u 3 ; u 1 = ; l 2 = u 3 ; u 2 = As a result, the end points of the curve x 1 = u 3 in M 2 are ( u3, and (, u3 It s easy to see that if x 1 and x 2 u3, the the right end point would be (, u3, and if x 1 u3 and, the right end point would be ( u3, So if x 1 and x 2 u3 and u1 x 1 l3, the plane (33 would touch the curve x 1 = u 3 at (, u3 and therefore we will have: (38 x 2( x 1+x 1( u 3 x 2+a(u 3 l 3 =0, and as a result (39 a = x 2( x 1+x 1( u3 x 2 l 3 u 3 On the other hand, if x 1 u3 and and x 1 u3, the plane (33 would touch the curve x 1 = u 3 at ( u3, and we will have: (310 x 2( u 3 x 1+x 1( x 2+a(u 3 l 3 =0,

5 hence, Inequalities for multilinear sets 17 (311 a = x 2( u3 x 1+x 1( x 2 l 3 u 3 (, l3 ( u3, ( u3, u2 ( u1, u3 (, u3 0 ( l3, x 1 Figure 2 Now we need to deal with two other cases; l3 x 1, x 2 = and x 1 =, l 3 x 2 First consider the case which l3 x 1, x 2 = Intuitively we can see that the plane which goes through the points ( l3,, l 3, (,,, and (, u3,u 3 gives an inequality which is valid for M 2 and is not dominated by any other inequality Similarly for the second case in which x 1 =, l3 x 2, the plane which goes through the points (, l3, l 3, (,,, and ( u3,,u 3 gives an inequality which is valid for M 2 and is not dominated by any other inequality u In summary, as can be seen in Figure 2, if 3 <x 1 < u1 u3 and <x 2 <, then the point ( x 1, (the point at which the plane (33 touches the curve x 1 = u 3 will have u3 < x 1 < and u3 < < Otherwise, ( x 1, happens at either ( u3, or(, u3 References 1 FA Al-Khayyal and JE Falk Jointly constrained biconvex programming Mathematics of Operations Research, 8: , GP McCormick Nonlinear programming: theory, algorithms and applications John Wiley & sons, EMB Smith and CC Pantelides Global optimisation of nonconvex MINLPs Computers & Chemical Engineering, 21:S791 S796, M Tawarmalani and NV Sahinidis Global optimization of mixed-integer nonlinear programs: A theoretical and computational study Mathematical Programming, 99(3: ,2004