By M. L. Balinski, Eli Hellerman
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Extra info for Computational Practice in Mathematical Programming
The signiﬁcance of this assertion is that if we know h then the maximization principle (M ) provides us with a formula for computing α∗ (·), or at least extracting useful information. We will see in the next chapter that assertion (M ) is a special case of the general Pontryagin Maximum Principle. Proof. 1. We know 0 ∈ ∂K(τ ∗ , x0 ). Since K(τ ∗ , x0 ) is convex, there exists a supporting plane to K(τ ∗ , x0 ) at 0; this means that for some g = 0, we have g · x1 ≤ 0 for all x1 ∈ K(τ ∗ , x0 ). 2.
1, where we could take any curve x(·) as a candidate for a minimizer. Now it is a general principle of variational and optimization theory that “constraints create Lagrange multipliers” and furthermore that these Lagrange multipliers often “contain valuable information”. This section provides a quick review of the standard method of Lagrange multipliers in solving multivariable constrained optimization problems. UNCONSTRAINED OPTIMIZATION. Suppose ﬁrst that we wish to ﬁnd a maximum point for a given smooth function f : Rn → R.
A|≤M So α(t) = M −M if q(t) < p2 (t) if q(t) > p2 (t) 64 for p2 (t) := λ(t − T ) + q(T ). CRITIQUE. In some situations the amount of money on hand x2 (·) becomes negative for part of the time. The economic problem has a natural constraint x2 ≥ 0 (unless we can borrow with no interest charges) which we did not take into account in the mathematical model. 7 MAXIMUM PRINCIPLE WITH STATE CONSTRAINTS We return once again to our usual setting: ˙ x(t) = f (x(t), α(t)) (ODE) x(0) = x0 , τ (P) P [α(·)] = r(x(t), α(t)) dt 0 for τ = τ [α(·)], the ﬁrst time that x(τ ) = x1 .
Computational Practice in Mathematical Programming by M. L. Balinski, Eli Hellerman