The uncertain parameters of the joint are assumed to be a convex set, hyper-rectangle or ellipsoid.
A real-valued function is defined to be a convex function if its epigraph is a convex set.
Importantly, the probability flow is a convex function of the parameters, consists of a number of terms linear in n and the size of X, and avoids the exponentially large partition function Z.
Under the previous assumption of perfect knowledge of the channel coefficient and the noise variance, the problem is a convex optimization problem with the parameter P RD i and P SR i.
From now on, it is assumed that ( X, d ) is a b-metric space (resp. ( X, d, W ) is a convex b-metric space) with parameter s and that S, T : Y → X are two nonself mappings on a subset Y of X such that T ( Y ) ⊂ S ( Y ), where S ( Y ) is a complete subspace of X.
Now we consider the following regularized minimization problem: min x ∈ C f α ( x ) : = f ( x ) + α 2 ∥ x ∥ 2, where α > 0 is the regularization parameter, f is a convex function with a 1 / L -ism continuous gradient ∇f.
Now, we consider the following regularized minimization problem: min_{xin C}g_{alpha}(x):=g(x)+frac{alpha}{2}|x |^{2}, where (alpha>0) is the regularization parameter, g is a convex function with a (1/L -ism continuous gradient ∇g.
Now, we consider the following regularized minimization problem: min_{xin C}g_{lambda}(x):=g(x)+frac{lambda}{2}|x |^{2}, where (lambda>0) is the regularization parameter, g is a convex function with a (1/L -ism continuous gradient ∇g.
This is a convex optimization problem, since the objective function is linear and the set of feasible u_i is convex.
The definition of a convex set implies that the intersection of two convex sets is a convex set.
