Optimality conditions for maximizing a function over a polyhedron

A General Regularized Continuous Formulation for the Maximum Clique Problem

“…Proof. The lemma follows immediately from [10, Corollary 2.2] and the fact that D is a reflective edge-description (defined in [10]) of ∆.…”

“…Next, note that by definition of D for each s ∈ [k] there exist i, j ∈ [n] such that d s = (e i − e j ); moreover, since (e i − e j ) ∈ F(x(S)), we have that x i ≤ In the proof of the next proposition, we will use the following well-known second-order sufficient optimality condition (see [10]): A point x ∈ ∆ is a local maximizer of (3) if…”

“…for all sufficiently small t > 0. So by the standard first-order necessary local optimality condition (see for example, Section 1 of [10]), we have ( 14)…”

“…In fact, by symmetry it is easy to see that the inequality in (26) must actually be an equality. In the proof of the next proposition, we will use the following well-known second-order sufficient optimality condition (see [10]…”

Continuous quadratic programming formulations of optimization problems on graphs

“…The next proposition will determine the amount by which we must decrease γ in order to ensure that the current point (x, y), a local maximizer of f γ , is no longer a local maximizer of the perturbed problem f γ . The derivation requires a formulation of the second-order necessary and sufficient optimality conditions given in [17,Cor. 3.3]; applying these conditions to the bilinear program (3.15), we have the following theorem.…”