We consider empirical risk minimization of linear predictors with convex loss functions. Such problems can be reformulated as convex-concave saddle point problems, and thus are well suitable for primal-dual first-order algorithms. However, primal-dual algorithms often require explicit strongly convex regularization in order to obtain fast linear convergence, and the required dual proximal mappi...

Existence analysis of primal and dual evolution mixed variational inclusions is performed on the basis of duality principles, rendering primal and dual solvability equivalence, respectively. Via a fixed-point maximal monotone subdifferential resolvent characterization, corresponding existence results are established under a strong monotonicity condition for the time derivative-elliptic combined...

We define the primal and dual linear programming problems involving interval numbers as the way of traditional linear programming problems. We discuss the solution concepts of primal and dual linear programming problems involving interval numbers without converting them to classical linear programming problems. By introducing new arithmetic operations between interval numbers, we prove the weak...

Interior-point methods possess strong theoretical properties and have been successfully applied to a wide variety of linear and nonlinear programming applications. This paper presents a class of algorithms, based on interior-point methodology, for performing regularized maximum likelihood reconstructions on 3-D emission tomography data. The algorithms solve a sequence of subproblems that conver...

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. ...

1.1 Complementary Slackness: Full and Approximate Recall our canonical linear programs, where x ∈ R, y ∈ R, A ∈ R, b ∈ R, c ∈ R. Primal (P) min cx Ax ≥ b x ≥ 0 Dual (D) max by A y ≤ c y ≥ 0 Also recall that Strong Duality ensures that if both (P) and (D) have finite optima, they are equal. Assume this is the case. Then, for the optima x and y of the primal and dual programs respectively, we hav...

Duality results on countably infinite linear programs are scarce. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix. Subspaces that permit a dual with this transpose do not admit an interior point. This difficulty has stumped researche...

In this paper, an exact dual is derived for Semideenite Programming (SDP), for which strong duality properties hold without any regularity assumptions. Its main features are: i. The new dual is an explicit semideenite program with polynomially many variables and polynomial size coeecient bitlengths. ii. If the primal is feasible, then it is bounded if and only if the dual is feasible. iii. When...

The generation of high-quality staggered unstructured grids for computational simulation is considered; leading to the development of a new optimisation-based strategy for the construction of weighted ‘RegularPower’ tessellations appropriate for co-volume type numerical techniques. This new framework aims to extend the conventional Delaunay-Voronoi primal-dual structure; seeking to assemble gen...

Quantified Boolean Formulas (QBF) provide a good language for modeling many complex questions about deterministic systems, especially questions involving control of such systems and optimizing choices. However, translators typically have one set way to encode the description of a system and a property, or question about the system, often without distinguishing between the two. In many cases the...