By Henry M. Paynter

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11. Consider two set of active constraints Aj and Aj and let CRi and CRj be the corresponding critical region, respectively. 34) is non-degenerate and that CR ¯ i and CR ¯ j denote the closure of the sets CRi and CRj , respectively. where CR Then, Ai ⊂ Aj and #Ai = #Aj − 1 or Aj ⊂ Ai and #Ai = #Aj + 1. 2. 10. This fact, together with the convexity of the set of feasible parameters K ∗ ⊆ K and the piecewise linearity of the solution z ∗ (x) is proved in the next Theorem. 12. 34) and let H 0. Then, the set of feasible parameters K ∗ ⊆ K is convex.

The function J ∗ : K ∗ → R will denote the function which expresses the dependence on x of the minimum value of the objective function over K ∗ , J ∗ (·) will be called value function. The set-valued function sc s Z ∗ : K ∗ → 2R × 2{0,1} d will describe for any fixed x ∈ K ∗ the set of ∗ optimizers z (x) related to J ∗ (x). We aim at determining the region K ∗ ⊆ K of feasible parameters x and at finding the expression of the value function J ∗ (x) and the expression an optimizer function z ∗ (x) ∈ Z ∗ (x).

38) where nT is the number of rows Ti of the matrix T . 34) is feasible for such an x0 . 34) is infeasible for all x in the interior of K. 34), in order to obtain the corresponding optimal solution z0 . 34). 10. Let H 0. Consider a combination of active constraints A0 , and assume that LICQ holds. Then, the optimal z ∗ and the associated vector of Lagrange multipliers λ∗ are uniquely defined affine functions of x over the critical region CR0 . 39a) λi (Gi z − Wi − Si x) = 0, i = 1, . . 39b) to obtain the complementary slackness condition λ∗ ( −G H −1 G λ∗ − W − Sx) = 0.

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Analysis and design engineering systems by Henry M. Paynter

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