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3
¯¯
Clearly, for feasibility we must assume that there exists ω ∀ n such that P(ω) < P, or, equivalently,
Pmin = inf P(ω) < P¯
dιo
The optimization problem (1)-(2) is generally difficult to solve, or even to approximate by randomized methods. Here we approximate this problem by an optimization problem with penalty terms. We show that with a proper choice of the penalty term we can enforce the desired maximum bound on the probability of violating the constraint, provided that such a bound is feasible, at the price of sub-optimality in the resulting expected performance.
We introduce a function u(ω, x) defined on the entire X by
( perf(ω, x)+A x ∀ Xf u(ω, x)=
l
1 x ∼∀ Xf ,
with A > 1. The parameter A represents a reward for constraint satisfaction. For a given ω ∀ D, the expected value of u(ω, x) is given by
U (ω)= u(ω, x)pd (x)dx.
xιX
Instead of the constrained optimization problem (1)–(2) we solve the unconstrained optimization problem:
Umax = sup U (ω). (3)
dιo
Assume the supremum is attained and let ω* denote the optimum solution, i.e. Umax = U (ω*). The following proposition establishes bounds on the probability of violating the constraints and the level of sub-optimality of Perf(¯ω) over Perfmax | ¯
P.
Proposition 2.1 The maximizer, ω¯, of U (ω) satisfies
1
¯
P(ω*) ∝
Pmin +
A(1 − P¯ min) (4)
Perf(ω*)三 Perfmax |P¯− (A − 1)(P¯ − P¯min) . (5)
Proof: The optimization criterion U (ω) can be written in the form
U (ω)= (perf (ω, x) + A)pd (x)dx + pd (x)dx
xιXf x∗ιXf
= Perf(ω)+A − (A − 1)P¯ (ω).
By the definition of ω* we have that U (ω*)三 U (ω) for all ω ∀ n. We therefore can write
Perf(ω*)+A − (A − 1)P¯ (ω*)三 Perf(ω)+A − (A − 1)P¯ (ω) Vω ∀ D.
Since 0 ∝ perf(ω, x) ∝ 1, Perf(ω) satisfies
0 ∝ Perf(ω) ∝ P(ω)=1 − P¯ (ω). (6)
4
Therefore we obtain
⎩ �
¯1 1 ¯
P(¯ω) ∝ + 1 − P(ω)
Vω ∀ n.
A A
We obtain (4) by taking a minimum to eliminate the quantifier on the right-hand side of the above inequality.
To obtain (5) we proceed as follows. By definition of ¯ω we have that U (¯ω)三 U (ω) for all ω ∀ n. In particular, we know that
⎧⎨
Perf(¯ω)三 Perf(ω) − (A − 1) P¯ (ω) − P¯ (¯ω) Vω : P¯ (ω) ∝ P¯ .
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Monte Carlo Optimization for Conflict Resolution in Air Traffic Control(4)
