By Coste M.
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Convex research is the calculus of inequalities whereas Convex Optimization is its program. research is inherently the area of the mathematician whereas Optimization belongs to the engineer. In layman's phrases, the mathematical technological know-how of Optimization is the learn of ways to make a good selection while faced with conflicting requisites.
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Bd for the chosen ordering. 3 The curve selection lemma The triangulation theorem allows one to give a short proof of the following. 3. 13 (Curve selection lemma) Let S ⊂ Rn be a semialgebraic set. Let x ∈ clos(S), x ∈ S. Then there exists a continuous semialgebraic mapping γ : [0, 1] → Rn such that γ(0) = x and γ((0, 1]) ⊂ S. Proof. Replacing S with its intersection with a ball with center x and radius 1, we can assume S bounded. Then clos(S) is a compact semialgebraic set. By the triangulation theorem, there is a ﬁnite simplicial complex K and a semialgebraic homeomorphism h : |K| → clos(S), such that x = h(a) for a vertex a of K and S is the union of some open simplices of K.
By the triangulation theorem, there is a ﬁnite simplicial complex K and a semialgebraic homeomorphism h : |K| → clos(S), such that x = h(a) for a vertex a of K and S is the union of some open simplices of K. In particular, since x is in the closure of S and not in S, there is a simplex σ of K whose a ◦ is a vertex, and such that h(σ) ⊂ S. Taking a linear parametrization of the segment joining a to the barycenter of σ, we obtain δ : [0, 1] → σ such that ◦ δ(0) = a and δ((0, 1]) ⊂σ. Then γ = h◦δ satisﬁes the property of the theorem.
D. algorithm described in Chapter 2 does not suﬃce to eliminate quantiﬁers. d. are adjacent to others, except for the cells in a cylinder. We do not know, in general, what happens when we pass from a cylinder to another. d. adapted to the sphere, it is not diﬃcult to determine the topology from the cell decomposition. The two functions on the disk x2 + y 2 < 1, whose graphs are the two open hemispheres, have an obvious extension by continuity on the closed disk. We show an example where this is not so.