bv_cvxbook_extra_exercises

# The velocity updates as vt1 1 vt hft bt t 0

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Unformatted text preview: also heavier, when we increase xi ; there is a tradeoﬀ between stiﬀness and weight. Our goal is to design the stiﬀest truss, subject to bounds on the bar cross-sectional areas and total truss weight: l ≤ xi ≤ u, i = 1, . . . , m, Wtot (x) ≤ W, where l, u, and W are given. You may assume that K (x) ≻ 0 for all feasible vectors x. To obtain a speciﬁc optimization problem, we must say how we will measure the stiﬀness, and what model of the loads we will use. (a) There are several ways to form a scalar measure of how stiﬀ a truss is, for a given load f . In this problem we will use the elastic stored energy 1 E (x, f ) = f T K (x)−1 f 2 to measure the stiﬀness. Maximizing stiﬀness corresponds to minimizing E (x, f ). Show that E (x, f ) is a convex function of x on {x | K (x) ≻ 0}. Hint. Use Schur complements to prove that the epigraph is a convex set. (b) We can consider several diﬀerent scenarios that reﬂect our knowledge about the possible loadings f that can occur. T...
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