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We now prove the Proposition 1 that gives the condition of compactness of sublevel set. Proof. Let B d (r) and S d−1 (r) denote the ball and sphere of radius r, centered at the origin. By affine transformation, we can assume that X * contains the origin O, X * ⊂ B d (1), and X * ∩ S d−1 (1) = φ. Then, we have that for ∀x ∈ S d−1 (1), (∇f (x), x) ≥ f (x) − f(More)
Difference of convex functions (DC) programming is an important approach to nonconvex optimization problems because these structures can be encountered in several fields. Effective optimization methods, called DC algorithms, have been developed in deterministic optimization literature. In machine learning, a lot of important learning problems such as the(More)
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