N ECESSARY AND S UFFICIENT O PTIMALITY C ONDITIONS
FOR
G ENERALIZED C ONVEX N ONSMOOTH
M ULTIOBJECTIVE O PTIMIZATION
Napsu Karmitsa
email: [email protected]
Marko M. Mäkelä
and
Ville-Pekka Eronen
Department of Mathematics
University of Turku, Finland
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
O UTLINE
• Introduction & Motivation
• Background
– Tools from nonsmooth analysis
– Generalized pseudo-convexity
– Generalized quasi-convexity
– Relations between convexities
• Necessary optimality conditions
• Sufficient optimality conditions
• Summary and remarks
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
I NTRODUCTION & M OTIVATION
Consider a general multiobjective optimization problem
(
minimize
{f1(x), . . . , fq (x)}
(P)
subject to
x ∈ S,
where fk : Rn → R for k = 1, . . . , q are locally Lipschitz continuous functions.
Definition: A vector x∗ is said to be a global Pareto optimum of (P), if there does
not exist x ∈ S such, that fk (x) ≤ fk (x∗) for all k = 1, . . . , q and fl (x) < fl (x∗)
for some l, and a global weak Pareto optimum of (P), if there does not exist x ∈ S
such, that fk (x) < fk (x∗) for all k = 1, . . . , q. Vector x∗ is a local (weak) Pareto
optimum of (P), if there exists δ > 0 such, that x∗ is a global (weak) Pareto
optimum on B(x∗; δ) ∩ S.
Convexity plays a crucial role in mathematical optimization theory and generalized convexities have proven to be main tools when constructing optimality
conditions, particularly sufficient conditions for optimality.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
T OOLS
FROM
N ONSMOOTH A NALYSIS
The function f : Rn → R is convex if for all x, y ∈ Rn and λ ∈ [0, 1] we have
f λx + (1 − λ)y ≤ λf (x) + (1 − λ)f (y).
A function is locally Lipschitz continuous at a point x ∈ Rn (LLC at a point x ∈
Rn) if there exist scalars K > 0 and δ > 0 such that
|f (y) − f (z)| ≤ Kky − zk
for all y, z ∈ B(x; δ),
where B(x; δ) ⊂ Rn is an open ball with center x and radius δ.
If function is LLC at every point x ∈ Rn, then it is called locally Lipschitz
continuous (LLC).
Both convex and smooth functions are always LLC.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
T OOLS
FROM
N ONSMOOTH A NALYSIS (C ONT.)
Definition: Let f : Rn → R be LLC at x ∈ S ⊆ Rn. The Clarke generalized
directional derivative of f at x in the direction of d ∈ Rn is defined by
f ◦(x; d) := lim sup
y→x
t↓0
f (y + td) − f (y)
t
and the Clarke subdifferential of f at x by
∂f (x) := {ξ ∈ Rn | f ◦(x; d) ≥ ξ T d for all d ∈ Rn}.
Each element ξ ∈ ∂f (x) is called a subgradient of f at x.
The Clarke generalized directional derivative f ◦(x; d) always exists for a LLC
function.
If f is smooth, then ∂f (x) reduces to ∂f (x) = {∇f (x)}.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
T OOLS
FROM
N ONSMOOTH A NALYSIS (C ONT.)
Definition: The function f : Rn → R is said to be subdifferentially regular at
x ∈ Rn if it is LLC at x and for all d ∈ Rn the classical directional derivative
f (x + td) − f (x)
′
f (x; d) = lim
t↓0
t
exists and f ′(x; d) = f ◦(x; d).
Convexity, as well as smoothness implies subdifferential regularity.
Theorem 1: Let f : Rn → R be LLC at x∗. If f attains its local minimum at x∗,
then
0 ∈ ∂f (x∗).
If, in addition, f is convex, then the above condition is also sufficient for x∗ to
be a global minimum.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
G ENERALIZED P SEUDO -C ONVEXITY
Definition: A smooth function f : Rn → R is pseudo-convex, if for all x, y ∈ Rn
f (y) < f (x)
implies
∇f (x)T (y − x) < 0.
A smooth pseudo-convex function f attains a global minimum at x∗, if and
only if ∇f (x∗) = 0 .
Definition: A function f : Rn → R is f ◦-pseudo-convex, if it is LLC and for all
x, y ∈ Rn
f (y) < f (x)
implies
f ◦(x; y − x) < 0.
A convex function is always f ◦-pseudo-convex.
Theorem 2: An f ◦-pseudo-convex f attains its global minimum at x∗, if and
only if
0 ∈ ∂f (x∗).
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
G ENERALIZED Q UASI -C ONVEXITY
Definition: A function f : Rn → R is quasi-convex, if for all x, y ∈ Rn and
λ ∈ [0, 1]
f (λx + (1 − λ)y) ≤ max {f (x), f (y)}.
Definition: A function f : Rn → R is f ◦-quasi-convex, if it is LLC and for all
x, y ∈ Rn
f (y) ≤ f (x)
implies
f ◦(x; y − x) ≤ 0.
Definition: A function f : Rn → R is f ◦-quasi-concave if −f is f ◦-quasi-convex.
Theorem 3: A function f : Rn → R is f ◦ -quasi-concave if it is LLC and for all
x, y ∈ Rn
f (y) ≤ f (x)
implies
f ◦(y; y − x) ≤ 0.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
R ELATIONS B ETWEEN P SEUDO - AND Q UASI - CONVEXITIES
Convex
1)
o
Pseudo−Convex
f −Pseudo−Convex
o
Quasi−Convex
f −Quasi−Convex
2)
1) demands continuous differentiability
2) demands local Lipschitz continuity and subdifferential regularity
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
P SEUDO - AND Q UASI - CONVEXITIES : E XAMPLES
Pseudo-convex:
f (x) = x + x3
(smooth but not convex)
f ◦ -Pseudo-convex:
f (x) = min{|x|, x2}
Quasi-convex:

|x|,
f (x) = 1,

x − 1,
(LLC, not convex nor pseudo-convex)
when x ≤ 1
when 1 < x ≤ 2 (LLC, not f ◦ -quasi-convex nor f ◦ -pseudo-convex)
when x > 2
f ◦ -Quasi-convex:
f (x) = x3
Napsu Karmitsa
(smooth, not pseudo-convex nor f ◦ -pseudo-convex)
MCDC, Jyväskylä, Finland, 2011
N ECESSARY O PTIMALITY C ONDITIONS
Consider problem (P) with inequality constraints:
(
minimize
{f1(x), . . . , fk (x)}
(2)
subject to
gi(x) ≤ 0 for all i = 1, . . . , m,
where also gi : Rn → R for i = 1, . . . , m are LLC functions.
Denoting the total constraint function by
g(x) := max {gi(x) | i = 1, . . . , m}
the problem (2) is a special case of (P), where S = {x ∈ Rn | g(x) ≤ 0}.
Definition: The problem (2) satisfies the Cottle constraint qualification, if either
g(x) < 0 or 0 ∈
/ ∂g(x) when x is feasible.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
N ECESSARY O PTIMALITY C ONDITIONS (C ONT.)
Theorem 4: Suppose that the problem (2) satisfies the Cottle constraint qualification. If x∗ is a local weak Pareto optimum of (2), then there exist Lagrange
multipliers λj ≥ 0 for all j = 1, . . . , k and µi ≥ 0 for all i = 1, . . . , m such that
Pk
∗
j=1 λj > 0, µi gi (x ) = 0, i = 1, . . . , m, and
0∈
k
X
j=1
λj ∂fj (x∗) +
m
X
µi∂gi(x∗).
i=1
Corollary: The condition of Theorem 4 is also necessary for x∗ to be a local
Pareto optimum of (2).
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
N ECESSARY O PTIMALITY C ONDITIONS (C ONT.)
Consider problem (P) with both inequality and equality constraints.


{f1(x), . . . , fq (x)}
minimize
(3)
subject to
gi(x) ≤ 0 for all i = 1, . . . , m,


hj (x) = 0 for all j = 1, . . . , p,
where all functions are LLC.
Definition: In problem (3) the CQ constraint qualification holds at a point x if
TS (x) = G∗(x) ∩ H0(x),
where S is the feasible set of problem (3)
TS (x) = {d ∈ Rn | there exist ti ↓ 0 and di → d with x + tidi ∈ S}
is the tangent cone of the set S at the point x,
G∗(x) = {d | gi◦(x; d) ≤ 0 for all i such that gi(x) = 0} , and
◦
H0(x) = d | hj (x; d) = 0 for all j ∈ P .
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
N ECESSARY O PTIMALITY C ONDITIONS (C ONT.)
Theorem 5: If x∗ is a local weak Pareto optimum of (3) and the CQ constraint
qualification holds at x∗, then there exist coefficients λk ≥ 0 for all k ∈ Q, µi ≥ 0
for all i = 1, . . . , m, νj+ ≥ 0 and νj− ≤ 0 for all j ∈ P such that
0∈
q
X
k=1
Pq
λk ∂fk (x∗) +
m
X
i=1
µi∂gi(x∗) +
p
X
νj+∂hj (x∗) +
j=1
∗
λ
>
0
and
µ
g
(x
) = 0 for all i = 1, . . . , m.
k
i
i
k=1
p
X
νj−∂hj (x∗),
j=1
Corollary: The condition of Theorem 5 is also necessary for x∗ to be a local
Pareto optimum of (3).
Remark: In problem (2) the Cottle constraint qualification and the CQ constraint qualification lead to same necessary conditions. In other words, if either of constraint qualifications is satisfied the conditions in Theorem 4 are the
necessary conditions for local weak Pareto optimum.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
S UFFICIENT O PTIMALITY C ONDITIONS
Theorem 6: Let x∗ be a feasible point of problem (3). Suppose there exist
Lagrange multipliers λk ≥ 0 for all k = 1, . . . , q, µi ≥ 0 for all i = 1, . . . , m and
νj ∈ R for all j = 1, . . . , p such that
0∈
q
X
k=1
Pq
λk ∂fk (x∗) +
m
X
i=1
µi∂gi(x∗) +
p
X
νj ∂hj (x∗),
j=1
∗
+
λ
>
0
and
µ
g
(x
)
=
0
for
i
=
1,
.
.
.
,
m.
Let
J
= {j | νj > 0} and
k
i
i
k=1
J − = {j | νj < 0}. If fk are f ◦ -pseudo-convex for all k = 1, . . . , q, gi are f ◦ quasi-convex for all i = 1, . . . , m, hj are f ◦ -quasi-convex for j ∈ J + and hj are f ◦
-quasi-concave for j ∈ J −, then x∗ is a global weak Pareto optimum of (3).
Corollary: The condition of Theorem 6 is also sufficient for x∗ to be a global
Pareto optimum of (3), if in addition λk > 0 for all k = 1, . . . , q.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
S UMMARY AND R EMARKS
• We have considered KKT type necessary and sufficient conditions for
nonsmooth multiobjective optimization. Both inequality and equality
constraints were considered. The optimality was characterized as weak
Pareto optimality.
• In necessary conditions the Cottle or CQ constraint qualifications were
needed.
• In sufficient conditions the main tools used were the generalized
pseudo- and quasi-convexities based on the Clarke generalized directional derivative. In inequality constrained case, it was assumed that all
the objective functions are f ◦-pseudo-convex and the constraint functions are f ◦-quasi-convex.
• Due to relations between different generalized convexities the previous
result is valid also for f ◦-pseudo-convex and subdifferentially regular
quasi-convex constraint functions.
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011
Thank You!
Napsu Karmitsa
MCDC, Jyväskylä, Finland, 2011