This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to … Read more
This paper has a two-fold focus on proving that the quasimetric and the weak $\tau$-distance versions of the Ekeland variational principle are equivalent in the sense that one implies the other and on presenting the need of such extensions for possible applications in the formation and break of workers hiring and firing routines. Article Download … Read more
Proximal splitting algorithms are becoming popular to solve convex optimization problems in variational image processing. Within this class, Douglas-Rachford (DR) and ADMM are designed to minimize the sum of two proper lower semicontinuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local convergence behaviour of … Read more
We suggest a modification of the descent splitting methods for decomposable composite optimization problems, which maintains the basic convergence properties, but enables one to reduce the computational expenses per iteration and to provide computations in a distributed manner. It consists in making coordinate-wise steps together with a special threshold control. Citation Kazan Federal University, Kazan … Read more
In this paper we study proximal conditional-gradient (CG) and proximal gradient-projection type algorithms for a block-structured constrained nonconvex optimization model, which arises naturally from tensor data analysis. First, we introduce a new notion of $\epsilon$-stationarity, which is suitable for the structured problem under consideration. %, compared with other similar solution concepts. We then propose two … Read more
We propose a forward-backward proximal-type algorithm with inertial/memory effects for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. The sequence of iterates generated by the algorithm converges to a critical point of the objective function provided an appropriate regularization of the objective satisfies the Kurdyka-Lojasiewicz inequality, which is … Read more
We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method … Read more
This paper considers computation of Fr\’echet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to … Read more
In this paper, we study differential properties of Euclidean projection onto the power cone $K^{(p,q)}_n=\{(x,y,z)\in \mathbb{R}_+\times \mathbb{R}_+\times \mathbb{R}^n,\norm{z} \leq x^p y^q\}$, where $0< p,q < 1, p+q=1$. Projections onto certain power cones are examples of semismooth but non-strongly-semismooth projection onto a convex cone. Citation Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang ... Read more
In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the L2−Lp minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an epsilon-KKT point within O(log(1/epsilon)) iterations. The same result is also … Read more