Lower subdifferentiability of quadratic functions

In this paper we characterize those quadratic functions whose restrictions to a convex set are boundedly lower subdifferentiable and, for the case of closed hyperbolic convex sets, those which are lower subdifferentiable but not boundedly lower subdifferentiable. Once characterized, we will study the applicability of the cutting plane algorithm of Plastria to problems where the objective function is quadratic and boundedly lower subdifferentiable. AMS Subject classification: 26B25, 90C20.


Introduction
The notion of lower subdifferentiability was introduced by Plastria (12], as a relaxation of the concept of subdifferentiability of convex analysis. The motivation for introducing this new notion was algorithmic, since Plastria proved that the classical cutting plane method of Kelley for convex optimization also works, under appropriate assumptions, using lower subgradients to generate the cutting planes [12,13]. He also observed that lower subdifferentiability of a function implies quasiconvexity. Later, it was shown [8,11] that the notion of lower subdifferentiability can be obtained as a particular case of the c-subdifferentiability of Balder [3], within the framework of the generalized conjugation theory of Moreau [10], and conditions were given under which a quasiconvex fuction is lower subdifferentiable [8]. In the latter paper, relations between the lower subdifferential, the tangential of Crouzeix [5] and the quasisubdifferential of Greenberg and Pierskalla [6] were studied. Some further results on lower subdifferentiable functions defined on locally convex spaces can be found in [9]; applications in the field of fractional programming are given in [4].
Based on the above mentioned results on lower subdifferentiability, we regard this conceptas a kind of qualified quasiconvexity which, on one hand, is not too restrictive and, on the other, provides a new tool in quasiconvex analysis which, in some aspects, plays a role similar to that played by the subgradient in convex analysis.
Quasiconvexity of quadratic functions has been investigated by several authors (see [16,1] and the references contained therein). Motivated by our belief that lower subdifferentiability is probably the appropriate condition one has to impose to quasiconvex functions in order to obtain a useful theory parallel, to some extent, to convex analysis, we address in this paper to the problem of finding conditions under which the restriction of a quadratic function to a convex domain is lower subdifferentiable. We rely upon the fundamental work of S. Schaible [16] characterizing quasiconvex quadratic functions. When using his results in the text, we refer to the recent book [1] on generalized concavity.
We shall use the following notation. By m. we shall denote the extended real line [-oo, +oo]; by m.+ the set of real nonnegative numbers; the Euclidean scalar product of x and y, vectors of m.n, will be denoted by zT y, where T indicates transposition. We shall denote by 11 • 11 the Euclidean norm and by B (O; N) the closed hall with radius N and center the origin. For nonnecessarily square matrices B we will consider the norm subordinate to the Euclidean vectorial norm, it is, where p denotes the spectral radius. H the matrix is not square, we will say that it is orthogonal if we have BT B = I, where I represents the identity matrix. We will use the notation diag ( a1, ... , an) to denote a diagonal matrix having a 1 , ... , ª" as its diagonal en tries.
The same symbol A is used to denote a real matrix m x n A and the corresponding linear transformation x---+ Ax from JR" to mm. For notions of convex analysis we will use the standard terminology and notation of [14], with the following exceptions: we will denote by co K and co K the convex and closed convex hull of the set K e JR", respectively. A convex set K is We will consider quadratic functions Q(x) = ½xT Ax+ bT x, where A is a n x n real symmetric matrix and b E JR".
Plastria [12} extended the notion of subdifferentiability as follows: The paper is organized as follows. In Section 2, we give sorne characterization of b.l.s.d. quadratic functions defined on convex domains. In Section 3, we state necessary conditions for lower subdifferentiability of quadratic functions which, for hyperbolic (in the sense of [2]) closed convex sets, turn out to be also sufficient. Finally, in Section 4 we study the specialization of the cutting plane algorithm of Plastria [12] to the case of a quadratic objective function.

Quadratic b.l.s.d. functions
In this section the following lemma will be useful: Lemma 2.1 Let K e R" be a convez set and let f : el K -+ R be a continuous /unction. Then the f ollowing statements are equivalent: 9. f is quasiconvex on el K.
Proof: Since the implications 3.=>2.=>1. are obvious, we only have to prove that 1.=>3 .. Let x, y be two points in el K and let .\ E [O, 1]. Since, by [14, p.46, th.6.3], el K = el (ri K), there exist two sequences (xn), (Yn) in ri K that converge to x, y respectively. By the continuity of f, the sequences (f(xn)), (/(yn)) converge to f(x), f(y) respectively. Since f is quasiconvex on ri K, we have Taking limits as n --+ oc, we have  f(x) = 1 for x E (O, 1)). However, it is easy to see that upper semicontinuity suffices for the above equivalences to hold in the case of functions of one real variable.
However, when n > 1, upper semicontinuity is not enough ( take, e.g., Since (x~) lies in a compact set, we can assume, without loss of generality, that it converges to some x• E 8(0; N).
We shall prove that x• E a-¡lclK(xo).
Let x E el K be such that / ( x) < f ( xo) ; then, beca use of the convergence of (xn) to xo and the continuity off, there exists no such that f(x) < f(xn) whence, taking the limit as n -+ oo, we obtain In the general case, we can write x = limn-00 x~, where x~ E ri K and, analogously, we can find nb such that f(x~) < f(xo) for every n ~ nb . By the preceding result, we have Given a set K e Rn, as in [8, p.218), we will denote by ~(K) the union of the projections of K onto the hyperplanes whose intersection with K is nonempty, that is,  By [1, p.173, th.6.2], A has exactly one negative eigenvalue A1. Let t1 be a unitary eigenvector of A associated with A 1 and let N be a Lipschitz constant of Q on K. Then 3.• 4. According to [1, Section 6.1), there exists a bijective affine transformation x = Py + v from m.n into itself such that the composite function can be written as G(y) = ½YT Ay+   Given any y E D , using again Cauchy-Schwarz 's inequality and the thus, all variables y,, i = 2, ... , r, are bounded on D. This proves that AD is bounded. Therefore, since and pT is nonsingular, we conclude that AK is bounded. 4.=>1. We shall first assume that K itself is bounded. By Lemma 2.2, it suffi.ces to prove that Q¡int where Nis a Lipschitz constant for Q on ~(K) . Let x E int K be such that Q(x) < Q(xo) and let x' be the projection of x onto the hyperplane defined which proves our assertion.
Let us now consider the general case. Take P, v, G, A, 6, r and D as in the proof of the implication 3.=>4. and let Ar be the matrix obtained by taking the first r rows and columns of A. Since Gis quasiconvex on D, it is easy to prove that the function Given xo E K, for uo = IIr(P-1 (xo -v)) we have uo E IIr(D); thus, using the definition of G and the relation G = g o IIr, we can easily see that Hence, there exists Xo E a-Q¡K(xo) that verifies 11 Xo 11 < 11 p-l 11 •N, which con eludes the proof. O Remark. The proof of implication 3.=> l. in the preceding theorem that would be obtained joining the proofs of the implications 3.=>4. and 4.=> l. would be rather involved. However, it is rather easy to prove the weaker statement that if Q is bounded below on K then Q¡int K is l.s.d .. Indeed, by [1, p.174, th.6.3], there exists an upper bound 6 for Q on K Multiplying by (c5 -Q(x 0 )) 1 1 2 + (c5 -Q(x)) 1 1 2 one gets, after simplication, 1 ( Hence, if Q(x) < Q(xo) and mis a lower bound of Q on K , one has 1 ( which proves that is a lower subgradient of Q¡intK at xo . It is easy to see that it is not restrictive to take L a compact convex set ineluded in aff K. As proved by J. Bair [2, p.183, prop.5), this is equivalent to saying that the barrier cone of K coincides with the polar of o+ (el K) (which is just the barrier cone of o+ ( el K)) or, also, that the barrier cone of K is closed.
The class of hyperbolic convex sets includes all unbounded polyhedral convex sets and convex eones (except {O}) as particular cases.
The following two results are easily proved: Proposition 2.5 Let K e JR" be a nonempty convex set, L e JR" a bounded set and D e JR" a closed convex cone such that K e L + D. Then For a general dornain, i.e. nonnecessarily solid, we have the following characterization of quadratic functions that are b.l.s.d.: Corollary 2.8 Let K e JR" be a convex set and let Q(x) = ½xT Ax+ bT x be merely quasiconvex on K . Then the following statements are equivalent: e. Q is Lipschitzian on K , 9. Q is bounded below on K ,

The orthogonal projection o/ AK onto aff K is bounded.
Proof: If dirn aff K = p, we can write aff K = h(JRP) , where h : JR.P -+ JR" is defined by h(y) = By+c, for sorne orthogonal rnatrix B and sorne e E JR".
We have that Consequently, the boundedness of BT ABh-1 (K) is equivalent to that of BBT A(Ke) (as B is orthogonal) and, therefore, to 4., since BBT is just the orthogonal projection rnapping onto the subspace parallel to aff K.
The characterizations of b.l.s.d. quadratic functions given in the preceding section provide, obviously, sufficient conditions for a quadratic function restricted to a convex domain to be 1.s.d.. In this section, we will obtain a necessary condition which, for hyperbolic domains, is also sufficient.
We will need the following result: We will make use of the following notation, taken from the field of multiobjective optimization theory (see, e.g., [15, p.33 (i.e., e (X, D) is the set of minimal points of X with respect to the compatible (with the linear structure) order relation whose nonnegative cone is D ).
For the sake of clarity, we will first consider the case when the domain is solid.
H AO+(elK) = {O}, this latter inelusion is the one in the statement, which coneludes the proof in this case.
In the case Ao+ ( el K) -::/= {O}, sin ce AO+ ( el K) is a convex con e contained in o+(elAK) = m.+s, we have AO+(elK) = o+(elAK). Let us now assume that xo E K satisfies sT xo = sT v . We have to prove that xo E é (el K, C). Suppose, a contrario, that there exists d E C \ {O} such that xod E el K. Without loss of generality, we can assume that Ad = s. H xod were a local minimum of Q on el K , by Lemmas 2.1 and 3.1, it would also be a global minimum, whence Q would be bounded below on K.
But this would contradict Theorem 2.4. Therefore, there exists a sequence x1e E el K converging to xod and satisfying <O, which is a contradiction. Thus, we must have x 0 E e ( el K, C), which concludes the proof of condition 2.
Let us now assume that K is a closed hyperbolic convex set. Then, we can reformulate condition l. as:  (1), as k-+ oo). If Az > X, we have ( ) Since x1c E int K n int Le int {x E K 1 >-z = O} e int E and VQ(x1) -=/ O, reasoning analogously as we have done to prove that the denominator in a(x) does not vanish, one gets

We have that
From this we easily deduce that a(x) is bounded above on {x E K I Az < X, Q(x) < Q(xo)}. Hence, we have proved that for every xo E K such that VQ(xo) -::/ O, a(x) is bounded above on the set {x E KI Q(x) < Q(xo)}, as we needed in order to show the lower subdifferentiability of QIK at x 0 • Now, let xo E K be such that VQ(xo) =O. We have VQ(x1) = -1 VQ(x), since VQ(x) is an affine mapping. Let x E K with Q(x) < Q(xo). For large enough k, Q(x) < Q(xk) and, therefore, we can write inequality (1) as whence, taking limits as k -. oo, Analogously to the preceding case, we have to verify that the expression and, as x E intK, by 2. we have (x-v)T s < O. Defining m, h, M and X analogously to the preceding case and using the same reasoning as we used there, we deduce that P(x) is bounded above on {x E K I A:,; > X, Q(x) < Q(xo)}.
It is easy to verify that, similarly to the preceding case, Q¡p is b.l.s.d., where F = co {x E K I A:,; < X}, and that P(x) is bounded above on {x E K I A:,;~ X, Q(x) < Q(xo) }, which concludes the proof. O Remarks. l. Since, in the preceding theorem, s is unique up to a multiplication by a positive scalar and C does not depend on s, the set 2. If K is not a closed hyperbolic convex set, conditions l. and 2. of This expression has to be true for any A > 2, but, when .x ~ +oo, we have a contradiction.
Note that, in the preceding example, we have that AK is a closed hyperbolic set, which indicates that, in Theorem 3.2, the hypothesis that K is hyperbolic can not be relaxed to AK hyperbolic.
For the case when K is nonnecessarily solid, one has the following generalization of Theorem 3.2. First let us see that conditions l. and 2. are necessary. Let Q¡K be l.s.d.; it is easy to prove that Q o h¡h-l(K) is l.s.d. Therefore, by Theorem 3.2, o+(clBT ABh-1 (K)) = JR+z for sorne z E JRP \{O}, orthogonal to o+(c1h-1 (K)), and, also, and w E JRP is any vector that satisfies BT ABw + BT(Ac + b) =O.
Since O= BT(A(Bw +e) which indicates that xd (/. el K ; in this way, we have seen that x E t ( el K, C) , which proves that conditions l. and 2. are necessary.
Let us see now that if K is a elosed hyperbolic convex set, then conditions l. and 2. are suffi.cient for Q¡K to be l.s.d. In this case, there exists a bounded set L, which we can suppose to be ineluded in aff K, such that Therefore, h-1 (K) is also a elosed hyperbolic set.
Let w E ffi.P with BT ABw + BT(Ac + b) = O and let v = h(w) E aff K. One has II(Av + b) = BBT(A(Bw +e)+ b) = O and, therefore, We want to see now that condition By+ e E é (K, C) is equivalent to Let y E ffi.P satisfying By+c E e(K, C). lt is easy to prove that y E h- 1 (K) and that, for e E A\ {O}= o+(h-1 (K)) \ker BT AB, Be E BO+(h-1 (K)) = o+ (K) . Moreover, e ft. ker BT AB = ker BBT AB = ker ITAB, it is, Be Í ker ITA . Hence Be E o+ ( K) \ ker ITA = C \ {O} and therefore h(y -e) = B(y -e)+ e= By+ e -Be i K, i.e., ye i h-1 (K), which means that y E ! (h-1 (K), ~). Thus we are under the conditions of Theorem 3.2, for Q oh and the solid closed hyperbolic convex set h-1 (K); hence, Q o h¡h-1(K) is l.s.d. Since we deduce that QIK is 1.s.d., which concludes the proof.
Remark. Condition 2. in the preceding corollary can be written as In this section we will consider that the convex sets to which we restrict our functions are solid. This does not mean loss of generality, since for any nonempty convex set K e m. n such that p = dim aff K, defining h, B and e as in the proof of Corollary 2.8, we have that h-1 (K) is a solid convex set and Q o hih-l(K) is b.1.s.d. and if fi is optima! for g on h-1 (K), h(fi) is optimal for Q on K and conversely.
We know that if K e ffi." is a solid compact convex set and Q(x) = ½ xT Ax + bT:,; is merely quasiconvex on K , for any :,; 0 E K such that N VQ(xo) # O, one has that II VQ(xo) II VQ(xo) E a-qlK(z0), where N is a Lipschitz constant of Q on the set ~(K) (see Section 2 and the proof of Theorem 2.4). If VQ(xo) =O, taking x E int K, we have II V~x) II VQ(x) E a-qlK(xo) (see the proof of Theorem 3.2). Note that, in the latter case, zo belongs to the boundary of K , since Q is merely pseudoconvex on int K (see [1, p.179, cor.6.4]). It is easy to see, using [1, p.174, th.6.3], that the following proposition holds: Proposition 4.1 Let K e 1R" be a nonempty compact convex set and let Q(x) = ½xT Ax+ bT x be merely quasiconvex on K. 1/ VQ(xo) =O, then xo is a maximum o/ Q on K .
We want to solve the problem where K is a solid convex polytope and Q a quadratic merely qua.siconvex function on K. By Theorem 2.4, these hypotheses imply that Q¡K is b.l.s.d.
We can transform problem (P) into problem Next we describe the cutting plane algorithm of Pla.stria [12, p.48] for minimizing b.l.s.d. functions on polytopes, specialized to the quadratic ca.se. We will denote by Na Lipschitz constant of Q on ~(K) and by e a fixed positive number.
Increase k by 1 and return to step 2).
The sequence (t1c)1c generated by the algorithm is non-decreasing, by definition of problems (P1r.), However, since we can not assure that sequence (Q(x1r.))1r. were non-decreasing, we have to consider the possibility that, in some iteration, we obtain VQ(x1c) = O. In this case, we can take XÁ: = II VQ~xo) II VQ(xo); when VQ(x1r.) :f O, x¡ = II v:x1r.) II VQ(x1r.) can be taken as a lower subgradient. In practice, if we are in the first case, instead of adding the constraint t ~ x¡T (x -x1c) + Q(xk) it is preferable to modify the constraint corresponding to j = O altering, if necessary, its independent term {since both linear inequalities determine parallel halfspaces).