By Christodoulos A. Floudas
Significant learn task has happened within the zone of worldwide optimization in recent times. Many new theoretical, algorithmic, and computational contributions have resulted. regardless of the foremost significance of try out difficulties for researchers, there was a scarcity of consultant nonconvex try difficulties for limited international optimization algorithms. This e-book is encouraged by way of the shortage of worldwide optimization try difficulties and represents the 1st systematic selection of attempt difficulties for comparing and trying out restricted international optimization algorithms. This assortment comprises difficulties coming up in quite a few engineering purposes, and attempt difficulties from released computational reports.
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Additional info for A Collection of Test Problems for Constrained Global Optimization Algorithms
P. Bertsekas, Nonlinear Programming, 2nd Edition, Athena Scientific, 2004. Further Reading 15 2. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. 3. M. L. Donoho, and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Review, 51(1):34– 81, February 2009. 4. S. L. A. Saunders, Atomic decomposition by basis pursuit, SIAM Journal on Scientific Computing, 20(1):33–61 (1998). 5. S. L. A. Saunders, Atomic decomposition by basis pursuit, SIAM Review, 43(1):129–159, 2001.
Clearly, though, an uncertainty result of the kind posed in Theorem 1 would be impossible to imitate, and we will have to bypass it somehow. 1 Uniqueness via the Spark A key property that is crucial for the study of uniqueness is the spark of the matrix A, a term coined and defined by Donoho and Elad in 2003. The spark is a way of characterizing the null-space of a matrix A using the 0 -norm. 2. : The spark of a given matrix A is the smallest number of columns from A that are linearly-dependent.
Equality is obtained if and only if x and y are linearly-dependent. 2 0 0 10 20 30 40 50 60 Fig. 1 The picket-fence signal for n = 64. It has 8 uniformly spread non-zeros with equal height. The Discrete-Fourier-Transform (DFT) of this signal looks exactly the same. eψ and eφ of the first n entries and last n entries of e, respectively. We have ΨeΨ = −ΦeΦ = y 0. 16) The vector y is nonzero because e is nonzero, and both Ψ and Φ are nonsingular. 4): e 0 = eΨ 0 + eΦ 0 ≥ 2 . 17) Since e = x1 − x2 , we have (Uncertainty Principle 2) : x1 0 + x2 0 ≥ e 0 ≥ 2 .
A Collection of Test Problems for Constrained Global Optimization Algorithms by Christodoulos A. Floudas