Mohammed-Najib Benbourhim, Patrick Chenin, Abdelhak Hassouni & Jean-Baptiste Hiriart-Urruty, Editors
ON PSEUDOCONVEX FUNCTIONS AND APPLICATIONS TO GLOBAL
OPTIMIZATION
A. HASSOUNI
1and A. JADDAR
1Abstract. In this paper, we characterize pseudoconvex functions using an abstract subdifferential. As applications, we also characterize maxima of pseudoconvex functions, and we study some fractional and quadratic optimization problems.
R´esum´e. Nous caract´erisons des fonctions pseudoconvexes en utilisant un sous diff´erentiel abstrait. Comme applications, nous caract´erisons ´egalement des maximums des fonctions pseudoconvexes , et nous ´etudions quelques probl`emes d’optimisation fractionnaires et quadratiques .
1.
Introduction
The pseudoconvexity notion that has been introduced first by Mangasarian in [17] has many applications in programming and mathematical economy. We will generalize some results of [4, 18, 19], where the authors have characterized a pseudoconvex function supposed to be radially continuous or radially non-constant. After recalling some preliminary results in section 2, we give in section 3 some results extending those of [1, 2, 4, 8, 19] for classes of functions that are less regular and where the assumptions of radial continuity and radial non-constancy are not always used. In section 4, we characterize maxima of pseudoconvex functions on convex sets. In section 5, we illustrate the theoretical results with two particular examples: a fractional and a quadratic problems.
2.
Some preliminary notions and results
In the sequel, byX we mean a Banach space andX∗ its dual for the duality pairingh. , .i. Forx∈X and
ε >0, we denote byBε(x) the “open” ball of centerxand radiusε. And forx, y∈X, the closed interval [x, y]
is the set n
tx+ (1−t)y: 0≤t≤1o.
For x6=y the semi-closed intervals (x, y],[x, y) and the open interval (x, y) are defined similarly by dropping one or two end-points. For anyA⊂X, we denote byint(A) its interior and by cl(A) its closure.
Let us recall that for any nonempty subsetC ofX and any pointxofX, the normal cone toC atxis defined by
N(C, x) = n
x∗∈X∗: ∀y∈C,hx∗, y−xi ≤0 o
.
1 D´epartement de math´ematiques et d’Informatique, Facult´e des Sciences, B.P. 1014, Rue Ibn Battouta,Rabat, Maroc; e-mail:[email protected] & [email protected]
c
°EDP Sciences, SMAI 2007
Consider a function f :X →IR∪ {+∞}, with a nonempty domain
domf = n
x∈X: f(x)<+∞ o
.
Forλ∈IR, the sublevel setSf(λ) is defined by
Sf(λ) = n
x∈X: f(x)≤λ
o
.
The functionf is said to be quasiconvex if for anyx, y∈X we have:
for anyz∈[x, y], f(z)≤maxnf(x), f(y)o.
And it is strictly quasiconvex if the above inequality is strict when x 6= y and z ∈ (x, y). The abstract subdifferential we consider here is defined as follows:
Definition 2.1. An operator∂ that associates with any l.s.c. function
f:X →IR∪ {+∞}and a pointx∈X a subset∂f(x)ofX∗ is a subdifferential if the following assertions hold :
(P1) ∂f(x) = n
x∗∈X∗: f(y)≥f(x) +hx∗, y−xi ∀y∈Xo
whenf is convex.
(P2) Ifx∈domf is a local minimum off, then 0∈∂f(x).
(P3) ∂f(x) =∂g(x), for any g:X →IR∪ {+∞}such that
(f−g)is constant in a neighborhood ofx.
(P4) ∂f(x) =∅, for anyx∈X such that f(x) = +∞.
In general, people working on the Mean Value Theorem know that to each kind∂of subdifferential corresponds a particular type of Banach space called∂-reliable, in which this Theorem is valid.
Definition 2.2. [18]A Banach spaceX is called∂-reliable if for each l.s.c. functionf:X →IR∪ {+∞}, for any Lipschitz convex function g and anyx∈domf such that f+g achieves its minimum in X atxand each
ε >0, we have:
0∈∂f(u) +∂g(v) +B∗ ε(0),
whereu, v ∈Bε(x)such that|f(u)−f(x)|< εandB∗
ε(0) is the “open” ball ofX∗ with center 0 and radiusε.
Indeed we have the fundamental result.
Theorem 2.1. [19] Let X be a ∂-reliable space and let f: X → IR∪ {+∞} be a l.s.c. function. For any
a, b ∈domf with a6= b, there exist a sequence (cn) in X converging to some c ∈ [a, b) and a sequence c∗ n in
∂f(cn)such that
i) lim inf n hc
∗
n, b−ai ≥f(b)−f(a).
ii) lim inf n
D
c∗ n,
||b−a||
||b−c||(b−cn) E
≥f(b)−f(a).
In the sequel we will use the “dag subdifferential”
∂†f(x) =nx∗∈X∗: hx∗, vi ≤f†(x, v) ∀v∈Xo,
where
f†(x, v) = lim sup
(t,y)→(0+,x)
t−1
³
f(y+t(v+x−y)−f(y) ´
It is a subdifferential introduced by Penot (see [18]) that is large enough to contain the Clarke-Rockafellar∂CR and the Upper Dini∂D+ subdifferentials and still has good properties.
Recall that an operator T:X →2X∗
is quasimonotone if for any x, y∈X the following implication holds ³
∃x∗∈T(x) : hx∗, y−xi>0 ´
=⇒ ³
∀y∗∈T(y) : hy∗, y−xi ≥0 ´
.
We have then the following relation between quasimonotonicity and quasiconvexity :
Theorem 2.2. [18, 19] Let X be a Banach space and let f:X →IR∪ {+∞} be a l.s.c. function. Consider the following assertions
i) f is quasiconvex.
ii) ∂f is quasimonotone. Theni) impliesii) if∂f ⊂∂†f.
Andii) impliesi) ifX is∂-reliable.
3.
Characterizations of pseudoconvex functions
In this section we study some properties of pseudoconvex functions.
Recall that a function f:X → IR∪ {+∞} is pseudoconvex for the subdifferential ∂ if for any x, y ∈ X the following implication holds
³
∃x∗∈∂f(x) : hx∗, y−xi ≥ 0´ =⇒ f(x)≤f(y).
The function f is strictly pseudoconvex if the right inequality that appears in the above implication is strict whenx6=y.
If C is an open convex set ofX, then we say thatf: C →IR∪ {+∞} is pseudoconvex (respectively strictly pseudoconvex) if the function defined by
ˆ
f(x) = ½
f(x) inC,
+∞ otherwise.
is pseudoconvex (respectively strictly pseudoconvex) onX.
We can easily check that a pseudoconvex functionf onX is pseudoconvex on any open convex subsetCofX. There is a close link between pseudoconvexity and quasiconvexity as we can see in the next result.
Theorem 3.1. Let X be a ∂-reliable space and let f: X → IR∪ {+∞} be a l.s.c. function. Consider the following assertions
i) f is pseudoconvex.
ii) f is quasiconvex and
³
0∈∂f(x) =⇒xis a global minimum off´.
Then, i) impliesii).
Andii) impliesi) iff is radially continuous and∂f ⊂∂†f.
Proof. The partii) ⇒ i) is similar to the proof of the corresponding assertion in [1] Theorem 7.1, so we prove only the parti) ⇒ii).
Indeed, by the very definition, it is sufficient to verify thatf is quasiconvex. If that was not the case, in view of the lower semicontinuity off, there would existx, y∈X, z∈(x, y) andε >0 such that
for allz0∈B
ε(z), f(z0)>max n
Since z cannot be a local minimum (because apparently z is not a global minimum and for a pseudoconvex function every local minimum is a global), there is some v in Bε(z) such that f(v) < f(z). Thanks to Theorem 2.1, there exist
(wn)→z¯∈[v, z) andw∗
n ∈∂f(wn) such that
hwn∗, z−wni>0.
But sincez∈(x, y), one of the two following cases must holds
hw∗
n, x−wni>0 or hw∗n, y−wni>0. Therefore
f(wn)≤max n
f(x), f(y) o
.
Hence contradiction follows. 2
When∂is the Clarke-Rockafellar subdifferential∂CR, [i)impliesii)] has been proved by Daniilidis-Hadjisavvas in [8].
In the particular case wheref is strictly pseudoconvex we have the following simplified form of Theorem 3.1.
Proposition 3.1. Let X be a ∂-reliable space and let f: X →IR∪ {+∞} be a l.s.c. function. Consider the following assertions
i) f is strictly pseudoconvex.
ii) f is strictly quasiconvex and
³
0∈∂f(x) =⇒xis a global minimum off´. Theni) impliesii) if∂f(x)is nonempty for any x∈X. Andii) impliesi) iff is radially continuous and∂f ⊂∂†f.
Proof. Letf be a strictly pseudoconvex function, then by Theorem 3.1, the functionf is quasiconvex and satisfies the following optimality condition
0∈∂f(x) =⇒³xis a global minimum off´.
According to Diewert [9], it suffices to prove thatf is radially non-constant. Assume by contradiction that there exists a closed segment [x, y] withx6=ywheref is constant. Letz∈(x, y) and apply the strict pseudoconvexity property to xandz, then
(f(z) =f(x)) =⇒ ³
∀z∗∈∂f(z) : hz∗, x−zi<0 ´
.
Using the same argument forz andy we obtain
(f(z) =f(y)) =⇒ ³∀z∗∈∂f(z) : hz∗, y−zi<0´.
Since∂f(z) is nonempty, it follows that
for allz∗∈∂f(z), hz∗, x−yi<0 and hz∗, x−yi>0.
A contradiction. Conversely, suppose thatf satisfies conditionii) of Proposition 3.1. Then by Theorem 3.1,f
is pseudoconvex. Suppose by contradiction that there existx6=y in X and x∗∈ ∂f(x) and y∗ ∈ ∂f(y) such that
It follows by pseudoconvexity that
for allz∈[x, y], f(z)≥f(x)≥f(y).
Sincef is quasiconvex, then we have
for allz∈[x, y], f(z) =f(x).
Sof is not radially non-constant onX (since f is constant on [x, y]).
Thenf is not strictly quasiconvex. 2
Geometrically we ca see that the subdifferential of a pseudoconvex functionf at any pointx∈X is a subset of the normal coneN
³
Sf(f(x)), x ´
to the sublevel setSf(f(x)), more precisely we have :
Proposition 3.2. Let X be a ∂-reliable space and f: X → IR∪ {+∞} be a function that is l.s.c. and pseudoconvex such that∂f ⊂∂†f. Then we have
for allx∈X, cl³IR+∂f(x) ´
⊂N³Sf(f(x)), x ´
.
Proof. Suppose for contradiction that there existsv such that
v∈cl
³
IR+∂f(x) ´
andv6∈N
³
Sf(f(x)), x ´
.
Without loss of generality we suppose thatv=x∗∈∂f(x). Then, we can find somey inSf(f(x)) andε >0 such that
for ally0∈Bε(y), hx∗, y0−xi>0.
Therefore we have:
for ally0∈Bε(y), f(y0)≥f(x)≥f(y). Thenf and by the pseudoconvexity off it is a global minimum of f . On the other hand, sincef†(x, y−x)>0, there exist (x
n)→xand (tn)→0+ such that
f(xn+tn(y−xn))> f(xn).
By Theorem 3.1,f is quasiconvex and then fornsufficiently large,
f(y)> f(xn), hence contradiction follows withy is a global minimum of f. 2 Now, recall that an operator T: X →2X∗
is said to be pseudomonotone if for any x, y∈X, the following implication holds:
³
∃x∗∈T(x) : hx∗, y−xi>0 ´
=⇒ ³
∀y∗∈T(y) : hy∗, y−xi>0 ´
.
The following characterization extends a similar one in [1, 4] to larger classes of subdifferentials and functions.
Theorem 3.2. Let X be a ∂-reliable space and letf: X →IR∪ {+∞} be a l.s.c. and pseudoconvex function with a convex domain such that ∂f ⊂∂†f. Consider the following assertions
i) f is pseudoconvex.
ii) ∂f is pseudomonotone. Then, i) impliesii).
Proof. The proof of the implicationii)⇒i)is similar to the proof of the corresponding assertion in ( [20] Theorem 4.1.)
For the implication i) ⇒ii), suppose for contradiction that there existx, y in X and x∗ ∈∂f(x),y∗ ∈∂f(y) such that
hx∗, y−xi>0 and hy∗, y−xi ≤0.
Then, from Proposition 3.2hx∗, y−xi>0 implies thatf(x)< f(y), and by the pseudonvexity off,hy∗, y−xi ≤0
implies thatf(y)≤f(x), hence a contradiction. 2
4.
Maxima of pseudoconvex functions
Consider the following maximization problem
(P) ½
maximize f(x),
subject to x∈C,
where f is supposed to be pseudoconvex, l.s.c. and radially continuous, andC is a nonempty convex set of
X. Forz∈C, denote by
Cz= n
x∈C: f(x) =f(z)o Then we have
Theorem 4.1. Let X be a ∂-reliable space and let f:X → IR∪ {+∞} be a l.s.c and pseudoconvex function such that for anyxinC, ∂f(x)is nonempty and ∂f(x)⊂∂†f(x). Letx¯∈C be such that
inf
C f < f(¯x).
Thenx¯ is a maximum of f onC if and only if
for allx∈Cx¯, ∂f(x)⊂N(C, x).
Proof. Suppose that
f(y)≤f(¯x) ∀y∈C, or C⊂Sf(f(¯x)). By Proposition 3.2 we have:
for allx∈Cx¯, ∂f(x)⊂N(Sf(f(x)), x)⊂N(C, x).
Conversely, assume for contradiction that there exists ¯z∈C such that
f(¯z)> f(¯x).
By hypothesis, we can find somez∈C withf(z)< f(¯x).
From the radial continuity off, there exists somex0∈(z,z¯) such that
f(x0) =f(¯x).
Sincez,z¯∈C and∂f(x0)⊂N(C, x0) we get
for allx∗
0∈∂f(x0), hx∗0, z−x0i = 0.
This result is a refinement of Theorem2.1 Hiriart-Urruty and Yu. S. Lediaev [15] where the function is supposed to be convex and continuous.
In the particular case where∂ is the Clarke-Rockafellar subdifferential∂CR and the setC takes the form
C = nx∈X : g(x)≤0o,
where g is supposed to be pseudoconvex and continuous, and such that there is somex0 ∈X with g(x0)<0,
we have
Proposition 4.1. Let X be a Banach space and f, g:X →IR∪ {+∞} be pseudoconvex and continuous with nonempty subdifferentials inC, and letx¯∈C be such that
inf
C f < f(¯x).
Thenx¯ is a maximum of f onC if and only if, for anyx∈C¯x
g(x) = 0 and ∂CRf(x)⊂cl³IR
+∂CRg(x) ´
.
Proof. Suppose that ¯xis a maximum off onC, let us first show that ¯xis on the boundary ofC. Assume by the contrary that ¯xis not on the boundary ofC. Therefore, there existx1, x2∈C such thatf(x1)< f(¯x)
and ¯x∈]x1, x2[. Sincef is pseudoconvex andf(x1)< f(¯x) then
hx¯∗, x
1−x¯i<0, ∀x¯∗∈∂CRf(¯x) (1)
From Proposition 3.2 andf(x2)≤f(¯x), we have
hx¯∗, x2−x¯i ≤0, ∀x¯∗∈∂CRf(¯x) (2)
As ¯x∈]x1, x2[, then (1) and (2) means that
hx¯∗, x1−x2i<0 andhx¯∗, x2−x1i ≤0, ∀x¯∗∈∂CRf(¯x)
Since ∂CRf(¯x) is nonempty, we get a contradiction. It follows that ¯xis on the boundary of C, hence g(¯x) = 0> g(x0), then by Theorem 4.1 and Proposition 2.2 of [12], we have that
for allx∈Cx¯, ∂CRf(x)⊂N(C, x) =cl ³
IR+∂CRg(x) ´
.
Conversely, since for anyx∈Cx¯, g(x) = 0, Proposition 2.2 of [12]
N(C, x) = cl
³
IR+∂f(x) ´
.
According to Theorem 4.1, ¯xis a maximum of f onC. 2
As an illustration of Theorem 4.1, we consider the following functionf: IR2→IR defined by
f(x, y) = ½
max{x, y} ifx <0 andy <0,
0 otherwise.
It is clear thatf is continuous and quasiconvex on IR2. Let us show that it is pseudoconvex on IR−
∗ ×IR−∗. Since for any (x, y)∈IR−∗ ×IR−∗ we have :
∂f(x, y) =
{(µ,1−µ) : µ∈[0,1]} forx=y,
{(0,1)} forx < y,
Then (0,0) 6∈ ∂f(x, y), by Theorem 3.1, f is pseudoconvex on the open convex set IR∗
−×IR∗−. Consider the convex setC= [−2,−1]2 and (−1,−1)∈C, then we can see that
C(−1,−1) = {(x,−1); (−1, y) : (x, y)∈C},
and that for any (x, y)∈C(−1,−1)
N(C,(x, y)) =
{0} ×IR+ for (x, y)∈]−2,−1[×{−1},
IR+× {0} for (x, y)∈ {−1}×]−2,−1[,
IR+×IR− for (x, y) = (−1,−2), IR−×IR+ for (x, y) = (−2,−1),
IR+×IR+ for (x, y) = (−1,−1),
So,∂f(x, y)⊂N(C,(x, y)). By Theorem 4.1, we conclude that (−1,−1) is a global maximum off onC.
5.
Applications to fractional or quadratic programming
First consider the following fractional problem
(P1) ½
maximizeq(x) = f(x)/g(x),
subject tox∈C.
Wheref andg are locally Lipschitz functions on some open convex setOcontaining the convex setC. Then the functionq is locally lipschitz. If we require in addition the following
C1) f is convex andg is concave onO,
C2) f is nonnegative andg is positive onO, thenqis pseudocopnvex on O, indeed, we have :
Proposition 5.1. Let X be a ∂-reliable Banach space and let q be a function defined as in (P1) such that
∂q⊂∂CRq. IfC1)andC2) hold, thenqis pseudoconvex on O.
Proof. For anyα∈IR, we observe that
Sq(α) = Shα(0),
where the function his defined by
hα(x) = f(x)−αg(x), ∀x∈O.
Sincehαis convex,q is quasiconvex.
In order to prove the pseudoconvexity of the functionq, it suffices to show that if 0∈∂q(x), thenxis necessarily a global minimum of the functionq.
Consider x∈O such that 0∈∂q(x), hence
0∈g(x)∂CRf(x)−f(x)∂CRg(x).
It follows that 0∈∂CRh
q(x)(x), this means thatxis a global minimum ofhq(x), and sincehq(x)(x) = 0, we have
f(y)−q(x)g(y)≥0, ∀y∈O.
As an illustration of Theorem 4.1, we study the following linear fractional problem :
(P2) ½
maximize f(x) = x1/(x1+x2),
subject to x= (x1, x2)∈C= [1,2]2.
By the preceding Proposition, the function f is pseudoconvex on X+, where
X+ = {x∈IR2: x1+x2>0}.
And since C is a compact subset of the half space X+, the maximum of f in C is achieved at some point
¯
x= (¯x1,x¯2).
On the other hand, for anyx∈C, we have
∇f(x) = 1 (x1+x2)2
µ
x2 −x1
¶
.
Then, it is easy to verify that the condition
∇f(x)∈N(C, x) ∀x∈C¯x,
holds for the point ¯x= (2,1), sinceCx¯={x¯}andN(C,x¯) = IR+×IR−. Then by Theorem 4.1, ¯xis a maximum
off onC.
And now consider the quadratic problem
(Q1) ½
maximize f(x) = 1
2hAx, xi+ha, xi+α,
subject to x∈C={x∈IRn: g(x)≤0},
whereg is a convex,g(x) = 12hBx, xi+hb, xi+β, where AandB are symmetric matrices.
Proposition 5.2. Let f andg be two functions defined as in(Q1)such thatgis convex andf is pseudoconvex on an open convex containingC. Considerx¯∈C such that the following assumptions hold:
i) There is some x0∈C such that g(x0)<0, ii) There is somey0∈C such thatf(y0)< f(¯x),
Then, x¯ is a solution of(Q1) if and only if for anyx∈Cx¯,
g(x) = 0 and ∃µ=µ(x)>0 such that Ax+a=µ(Bx+b).
The proof is omitted because this Proposition is only another way of stating Proposition 4.1. As an illustration, consider
(Q2) ½
maximize f(x) = −x1x2
subject to x= (x1, x2)∈C={x∈IRn : g(x)≤0},
where g(x) = 9
4x21+94x22−72x1x2−2x1−2x2+ 3.
Thenf andg are quadratic functions with associated matrices
A= µ
0 −1
−1 0 ¶
and B= µ
9/2 −7/2
−7/2 9/2 ¶
SinceB is positive semi-definite,g is convex.
On the other hand,f is pseudoconvex on the open convex set
O = {x= (x1, x2) : x1>0, x2>0}.
Indeed, since the sublevel sets off are convex, f is quasiconvex on the open convex setO with∇f(x)6= 0 for anyx∈O, then by Theorem 3.1,f is pseudoconvex onO.
Moreover, one can also verify thatg(x)≤0 if and only if
X2
2 + 4Y
2≤1,
where X = √ 2 2 ³
x1+x2−4 ´
and Y =
√
2 2
³
x2−x1 ´
.
It follows then thatC is included in the the disk of center (2,2) and radius√2 and then inD. Notice however thatf is pseudoconvex and not convex onC as we can check easily with the three points (1,1) (2,2) and (3,3). Consider the point ¯x= (¯x1,x¯2) = (1,1) it satisfies
µ
0 −1
−1 0 ¶ µ ¯ x1 ¯ x2 ¶
= ¯µ
µ
9/2 −7/2
−7/2 9/2 ¶ µ ¯ x1 ¯ x2 ¶ − µ 2¯µ 2¯µ ¶
with ¯µ= 1>0; moreover we can check easily thatCx¯={x¯}. Then ¯xis the solution of (Q2). 2
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