On dual convergence of the generalized proximal point method with Bregman distances (Q2757655)

The article under review is a valuable contribution to the algorithmical treatment of linearly constrained minimization of a differentiable convex function \(f\) in \(\mathbb{R}^n\). Here, the constraints are of the form \(Ax=b\), \(x\geq 0\). First, the authors recall the primal numerical concept, where the nonnegativity condition is taken into account by (a multiplier times) the Bregman distance \(D_\varphi\) of a separable convex barrier function \(\varphi\) added into the objective function, and by inserting \(x^k\) in order to obtain \(x^{k+1}\) as the argmin. However, they mainly refer to a broad class of separable barrier functions \(h\) (instead of \(D_\varphi)\). Likewise, fixing some inserted \(x^1\) gives rise to the primal central path \(x(\mu)\) of solutions depending on the multiplier variable, and cluster points of sequences \((x(\mu+ k))\). Then, they elaborate the dual concept given by the Lagrangian dual problem of maximizing \(\psi(s): =\inf\{f(x)- x^Ts\mid Ax=b\}\) subject to \(s\geq 0\). Numerous assumptions are made therefore, in particular: nonemptyness of primal solution and feasible sets, the \(n\) strictly convex additive functions \(h_j\) (defining \(h)\) having descent tending to \(-\infty\) when the variable tends to 0, \(h\) having a stationary point, and alternative conditions on the primal problem or on \(h\). Now, the dual central paths \(s(\mu)\) is defined as \(-\mu \nabla h(x(\mu))\), and it is shown to the unique solution of an auxiliary convex optimization problem, where the objective function is defined with the help of the conjugate convex functions of \(h\), \(h_j\), under an additional assumption imposed thereon. Cluster points of sequences \((s^k)\), corresponding to positive \(\mu_k \to 0\), turn out to be elements of the dual solution set \(S^*\). Under another assumption on decreasing (index set controlled) subsets of \(S^*\), which finally consist of a singleton called \(h\)-center \(s^c\) of \(S^*\), \(s(\mu)\) converges to \(s^c\).NEWLINENEWLINENEWLINEBesides the various auxiliary or main results, examples of barriers are presented (Bregman or divergence type), final remarks given, e.g., on the case of linear programming, and three open problems mentioned. The article is rich and carefully written.