In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems. We first discuss the semismoothness of the Lagrangian-dual function φ. This property is then used to investigate the second-order properties of the Moreau-Yosida regularization η of the function φ, e.g., the semismoothness of the gradient g of the regularized function η. We show that φ and g are p...

A subset S of R is said to be polyhedral if it is the intersection of a finite number of closed halfspaces, i.e., if there exist J ∈ N and collections {y1, . . . , yJ} ⊂ R, {α1, . . . , αJ} ⊂ R such that S = ∩j=1{x ∈ R : y jx ≤ αj}. A function f : R → [−∞,+∞] is polyhedral if its epigraph epi f ⊂ R is a polyhedral set. Clearly, any polyhedral set is automatically convex and closed. Consequently...

This paper is concerned with a constrained optimization problem over a directed graph (digraph) of nodes, in which the cost function is a sum of local objectives, and each node only knows its local objective and constraints. To collaboratively solve the optimization, most of the existing works require the interaction graph to be balanced or “doubly-stochastic”, which is quite restrictive and no...

Motivated by modern regression applications, in this paper, we study the convexification of a class convex optimization problems with indicator variables and combinatorial constraints on indicators. Unlike most previous work sparse problems, simultaneously consider nonlinear non-separable objective, variables, constraints. Specifically, give hull description epigraph composition one-dimensional...

STABLE AND TOTAL FENCHEL DUALITY FOR CONVEX OPTIMIZATION PROBLEMS IN LOCALLY CONVEX SPACES∗ CHONG LI† , DONGHUI FANG‡ , GENARO LÓPEZ§ , AND MARCO A. LÓPEZ¶ Abstract. We consider the optimization problem (PA) infx∈X{f(x) + g(Ax)} where f and g are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y , respectively, and A is a linear operator from X to Y ....

A biased Newton direction is introduced for minimizing quasiconvex functions with bounded level sets. It is a generalization of the usual Newton's direction for strictly convex quadratic functions. This new direction can be derived from the intersection of approximating hyperplanes to the epigraph at points on the boundary of the same level set. Based on that direction, an unconstrained minimiz...

In this paper, we study a general minimization vector problem which is expressed in terms of perturbation mapping defined on product locally convex Hausdorff topological spaces with values another space. Several representations the epigraph conjugate are given, and then, variants Farkas lemmas associated system by established. A dual so-called loose mentioned stable strong duality results betwe...

We establish two Phragmén–Lindelöf theorems for a fully nonlinear elliptic equation. consider dynamical boundary condition that includes both spatial variable and time derivative terms. As term, we non-linear Neumann-type operator with strict monotonicity in the normal direction of on term. Our first result is an equation epigraph $$\mathbb {R}^n$$ . Because assume good structural condition, wh...

A famous story about the Beatles tells of collaboration between Paul McCartney and John Lennon on song “Getting Better” for their legendary Sgt. Pepper's Lonely Hearts Club Band album. After wrote lines “I've got to admit, it's getting better; a little better all time,” wryly added, “It can't get no worse.” This could serve as an epigraph calendar turns from year 2020, which hardly have gotten ...

Among the rare materials of Department Manuscripts National Library France (Bibliotheque nationale de France. Departement des Manuscrits. Supplement turc 515) there is a small chronicle anonymous author dating back to era Crimean Khanate. The manuscript provided with description: “The history khans from 880 1191 Hegira (1475–1777), without title and name author. epigraph read on front side firs...