4.7 Vector optimization 175
objective function. We denote a general vector optimization problem as
minimize (with respect to K) f0(x)
subject to fi(x) ≤ 0, i = 1, . . . ,m
hi(x) = 0, i = 1, . . . , p.
(4.56)
Here x ∈ Rn is the optimization variable, K ⊆ Rq is a proper cone, f0 : Rn → Rq
is the objective function, fi : Rn → R are the inequality constraint functions, and
hi : Rn → R are the equality constraint functions. The only difference between this
problem and the standard optimization problem (4.1) is that here, the objective
function takes values in Rq, and the problem specification includes a proper cone
K, which is used to compare objective values. In the context of vector optimization,
the standard optimization problem (4.1) is sometimes called a scalar optimization
problem.
We say the vector optimization problem (4.56) is a convex vector optimization
problem if the objective function f0 isK-convex, the inequality constraint functions
f1, . . . , fm are convex, and the equality constraint functions h1, . . . , hp are affine.
(As in the scalar case, we usually express the equality constraints as Ax = b, where
A ∈ Rp×n.)
What meaning can we give to the vector optimization problem (4.56)? Suppose
x and y are two feasible points (i.e., they satisfy the constraints). Their associated
objective values, f0(x) and f0(y), are to be compared using the generalized inequal-
ity K . We interpret f0(x) K f0(y) as meaning that x is ‘better than or equal’ in
value to y (as judged by the objective f0, with respect to K). The confusing aspect
of vector optimization is that the two objective values f0(x) and f0(y) need not be
comparable; we can have neither f0(x) K f0(y) nor f0(y) K f0(x), i.e., neither
is better than the other. This cannot happen in a scalar objective optimization
problem.
4.7.2 Optimal points and values
We first consider a special case, in which the meaning of the vector optimization
problem is clear. Consider the set of objective values of feasible points,
O = {f0(x) | ∃x ∈ D, fi(x) ≤ 0, i = 1, . . . ,m, hi(x) = 0, i = 1, . . . , p} ⊆ Rq,
which is called the set of achievable objective values. If this set has a minimum
element (see §2.4.2), i.e., there is a feasible x such that f0(x) K f0(y) for all
feasible y, then we say x is optimal for the problem (4.56), and refer to f0(x) as
the optimal value of the problem. (When a vector optimization problem has an
optimal value, it is unique.) If x? is an optimal point, then f0(x?), the objective
at x?, can be compared to the objective at every other feasible point, and is better
than or equal to it. Roughly speaking, x? is unambiguously a best choice for x,
among feasible points.
A point x? is optimal if and only if it is feasible and
O ⊆ f0(x?) +K (4.57)

Part II
Applications