论robust优化理论与广泛应用：计算吸引力与跨领域实践

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Here x ∈ R

is a vector of decision variables, f

, f

: R

→ R are functions, and the uncertainty

parameters u

∈ R

are assumed to take arbitrary values in the uncertainty sets U

⊆ R

, which, for

our purposes, will always be closed. The goal of (2.3) is to compute minimum cost solutions x

∗

among

all those solutions which are feasible for all realizations of the disturbances u

within U

. Thus, if some

of the U

are continuous sets, (2.3), as stated, has an inﬁnite number of constraints. Intuitively, this

problem oﬀers some measure of feasibility protection for optimization problems containing parameters

which are not known exactly.

It is worthwhile to notice the following, straightforward facts about the problem statement of (2.3):

• The fact that the objective function is unaﬀected by parameter uncertainty is without loss of

generality; we may always introduce an auxiliary variable, call it t, and minimize t subject to the

additional constraint max

∈U

(x, u

) ≤ t.

• It is also without loss of generality to assume that the uncertainty set U has the form U =

× . . . × U

. If we have a single uncertainty set U for which we require (u

, . . . , u

) ∈ U, then

the constraint-wise feasibility requirement implies an equivalent problem is (2.3) with the U

taken

as the projection of U along the corresponding dimensions (see Ben-Tal and Nemirovski, [14] for

more on this).

• Constraints without uncertainty are also captured in this framework by assuming the correspond-

ing U

to be singletons.

• Problem (2.3) also contains the instances when the decision or disturbance vectors are contained

in more general vector spaces than R

or R

(e.g., S

in the case of semideﬁnite optimization)

with the deﬁnitions modiﬁed accordingly.

Robust Optimization is distinctly diﬀerent than sensitivity analysis, which is typically applied as a

post-optimization tool for quantifying the change in cost of the associated optimal solution with small

perturbations in the underlying problem data. Here, our goal is to compute ﬁxed solutions that ensure

feasibility independent of the data. In other words, such solutions have a priori ensured feasibility when

the problem parameters vary within the prescribed uncertainty set, which may be large. We refer the

reader to some of the standard optimization literature (e.g., Bertsimas and Tsitsiklis, [37], Boyd and

Vandenberghe, [43]) and works on perturbation theory (e.g., Freund, [75], Renegar, [123]) for more on

sensitivity analysis.

It is not at all clear when (2.3) is eﬃciently solvable. One might imagine that the addition of robust-

ness to a general optimization problem comes at the expense of signiﬁcantly increased computational

complexity. It turns out that this is true: the robust counterpart to an arbitrary convex optimization

problem is in general intractable ([13]; some approximation results for robust convex problems with

a conic structure are discussed in [35]). Despite this, there are many robust problems that may be

handled in a tractable manner, and much of the literature has focused on specifying classes of functions

, coupled with the types of uncertainty sets U

, that yield tractable robust counterparts. If we deﬁne

the robust feasible set to be

X(U) = {x | f

(x, u

) ≤ 0 ∀ u

∈ U

, i = 1, . . . , m}, (2.4)

then for the most part,

tractability is tantamount to X(U) being convex in x, with an eﬃciently

computable separation test. More precisely, in the next section we show that this is related to the

structure of a particular subproblem. We now present an abridged taxonomy of some of the main

results related to this issue.

2.2 Robust linear optimization

The robust counterpart of a linear optimization problem is written, without loss of generality, as

minimize c

subject to Ax ≤ b, ∀ a

∈ U

, . . . , a

∈ U

, (2.5)

where a

represents the i

row of the uncertain matrix A, and takes values in the uncertainty set

⊆ R

. Then, a

x ≤ b

, ∀a

∈ U

, if and only if max

∈U

}

x ≤ b

, ∀i. We refer to this as the

subproblem which must be solved. Ben-Tal and Nemirovski [14] show that the robust LP is essentially

always tractable for most practical uncertainty sets of interest. Of course, the resulting robust problem

may no longer be an LP. We now provide some more detailed examples.

Ellipsoidal Uncertainty: Ben-Tal and Nemirovski [14], as well as El Ghaoui et al. [77, 80],

consider ellipsoidal uncertainty sets. Controlling the size of these ellipsoidal sets, as in the theorem

below, has the interpretation of a budget of uncertainty that the decision-maker selects in order to

easily trade oﬀ robustness and performance.

i.e., subject to a Slater condition.

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论robust优化理论与广泛应用：计算吸引力与跨领域实践

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