We then consider a case of the problem (P) where there are no inequality constraints: (P0) min
x2Rn⇧(x). (3.36)
A more special problem of (P0) has been addressed in [46, 56]. It is a problem of
minimizing a forth-order (or quartic) polynomial,
min x2Rn 1 2 m X k=1 k( 1 2x TAkx ck)2+ 1 2x TQx xTf in which Q, Ak 2 Sn, f 2 Rn,
k 2 R++ and ck 2 R. The objective function
is a discretized form of the so-called double-well potential, first proposed by van der Waals in thermodynamics in 1895 [108], which is the mathematical model for natural phenomena of bifurcation and phase transitions in areas such as cosmology, continuum mechanics, material science, and quantum field theory [57, 71, 75].
As there are no inequality constraints in the problem (P0), the variables and &i
for i = 1, . . . , m will disappear from the dual problem and hence & = &0 2 Rp. The
equation (3.33) then becomes
G(&)x f (&) = 0, (3.37) with G(&) = A0 0+ m X k=1
&kA0k and f (&) = b00+ m
X
k=1
&kb0k.
The dual function then becomes
⇧d(&) = 1 2f (&)
TG(&) 1f (&) V⇤
0(&) (3.38)
with the feasible region
Sa={& 2 Ea⇤| det(G(&)) 6= 0}.
In (3.38), if G(&) 0, the first item of the dual function ⇧d will be concave in
the neighborhood of &. Combining with the convexity of V⇤
0(&), the dual function
⇧d will be concave in the neighborhood. Thus, we have the following result about
the convexity of ⇧d.
Lemma 6 If
Sc+ ={& 2 Sa| G(&) ⌫ 0}
is convex, the dual function ⇧d is a concave function on S+ c .
Before we present the following result, another subregion in Sa is introduced
Theorem 7 (Triality Theorem) Suppose that ¯& is a critical point of the dual function ⇧d(&) in (3.38) and ¯x = G( ¯&) 1f ( ¯&).
1. (min-max duality) If ¯& 2 S+
c , then ¯& is a global maximizer of ⇧d(&) over Sc+
and ¯x is a global minimizer of ⇧(x); moreover, the following equalities hold ⇧( ¯x) = min
x2Rn⇧(x) = max
&2Sc+
⇧d(&) = ⇧d( ¯&). (3.39)
2. (double-max duality) If ¯& 2 Sc , then ¯& is a local maximizer of ⇧d(&) if and
only if ¯x is a local maximizer of ⇧(x); there exists a neighborhood of ( ¯x, ¯&), Xo⇥ So ⇢ Rn⇥ Sc , such that ⇧( ¯x) = max x2Xo ⇧(x) = max &2So ⇧d(&) = ⇧d( ¯&). (3.40)
3. (double-min duality) If ¯& 2 Sc , we have the following cases:
(a) If p = n, then ¯& is a local minimizer of ⇧d(&) if and only if ¯x is a
local minimizer of ⇧(x); there exists a neighborhood of ( ¯x, ¯&), Xo⇥ So ⇢
Rn⇥ S c , such that ⇧( ¯x) = min x2Xo ⇧(x) = min &2So ⇧d(&) = ⇧d( ¯&). (3.41)
(b) If p < n, then ¯& is a local minimizer or a saddle point of ⇧d(&) if and only
if ¯x is a saddle point of ⇧(x);
(c) If p > n, then ¯& is a saddle point of ⇧d(&) if and only if ¯x is a local
minimizer or a saddle point of ⇧(x).
Proof: If ¯& 2 S+
c , from the global optimality in Theorem 3, it is true that ¯& is a
global maximizer of ⇧d(&) over S+
c , ¯x = G( ¯&) 1f ( ¯&) is a global minimizer of (P0)
and equalities in (3.39) hold.
In the rest of this proof, we assume that ¯& is a critical point of ⇧d(&) inS
c . Since
¯
& is a critical point, we have
0 = @⇧ d( ¯&) @&k = 1 2f ( ¯&) TG( ¯&) 1A0
kG( ¯&) 1f ( ¯&) bk0TG( ¯&) 1f ( ¯&) c0k
@V⇤ 0( ¯&)
@&k
,
for k = 1, . . . , p. By substituting G( ¯&) 1f ( ¯&) with ¯x, we then have
rV⇤ 0( ¯&) = ⇢ 1 2x¯ TA0 kx¯ x¯Tb0k p k=1 = ⇤0( ¯x), (3.42) which, by (3.3), is equivalent to ¯ & =rV0( ¯⇠) (3.43)
where ¯⇠ = ⇤0( ¯x). Hessian matrices of ⇧(x) and ⇧d(&) at ¯x and ¯& can then be
written as
r2⇧( ¯x) = G( ¯&) +r⇤
0( ¯x)r2V0( ¯⇠)r⇤0( ¯x)T, (3.44)
r2⇧d( ¯&) = r2V0⇤( ¯&) r⇤0( ¯x)TG( ¯&) 1r⇤0( ¯x). (3.45)
Then Lemma 44 and the fact of r2V⇤
0( ¯&) = (r2V0( ¯⇠)) 1 show that, by letting
P = G( ¯&), U =r2V
0( ¯⇠) and D =r⇤0( ¯x), we have the following equivalence
r2⇧( ¯x) 0, r2⇧d( ¯&) 0. (3.46) Thus, it is proved that ¯& being a local maximizer of (Pd
0) is equivalent to ¯x being a
local maximizer of (P0).
Then we prove the double-min duality. From the expressions in (3.44) and (3.45), we know that only when the second items in r2⇧( ¯x) and r2⇧d( ¯&) are positive
definite, the conditions r2⇧( ¯x)⌫ 0 and r2⇧d( ¯&) ⌫ 0 may hold true. Thus, when
the condition r2⇧( ¯x) ⌫ 0 is true, it can be proved that r2V
0( ¯⇠) is invertible and
rank(r⇤0( ¯x)) = p. When p = n, the matrix r⇤0( ¯x) is also invertible. By letting
P = r⇤0( ¯x)r2V0( ¯⇠)r⇤0( ¯x)T, U = G( ¯&) and D = I in Lemma 44, the following
equivalent relation can be derived
r⇤0( ¯x)r2V0( ¯⇠)r⇤0( ¯x)T + G( ¯&)⌫ 0
() G(¯&) 1 r⇤0( ¯x)r2V0( ¯⇠)r⇤0( ¯x)T 1
⌫ 0. (3.47) The positive semidefiniteness on the right side is further equivalent to
r⇤0( ¯x)TG( ¯&) 1r⇤0( ¯x) r2V0( ¯⇠) 1 ⌫ 0,
of which the left side isr2⇧d( ¯&), as r2V
0( ¯⇠) 1 =r2V0⇤( ¯&). Thus, it is proved that
when p = n, ¯& being a local minimizer of (Pd
p) is equivalent to ¯x being a local
minimizer of (Pp)
If the second items in r2⇧( ¯x) and r2⇧d( ¯&) are positive definite, we have the
following inequalities about the rank,
n = rank(r⇤0( ¯x)r2V0( ¯⇠)r⇤0( ¯x)T) rank(r⇤0( ¯x)) min{n, p} (3.48)
p = rank(r⇤0( ¯x)TG( ¯&) 1r⇤0( ¯x)) rank(r⇤0( ¯x)) min{n, p}. (3.49)
Thus, when p < n, the Hessian r2⇧( ¯x) can not be positive semidefinite, otherwise,
as mentioned above, it must be true that rank(r⇤0( ¯x)) = p, which will cause con-
tradiction in (3.48). Hence, ¯x can not be a local minimizer when p < n. Combining with the double-max duality, we have proved that ¯& is a local minimizer or a saddle point of ⇧d(&) if and only if ¯x is a saddle point of ⇧(x). Similarly, when p > n, we
can prove that ¯& is a saddle point of ⇧d(&) if and only if ¯x is a local minimizer or a
saddle point of ⇧(x).
Combining with Theorem 1, equations (3.39), (3.41) and (3.41) are also proved. 2