We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, O (1/N 2) with respect to initialisation andO (1/N ) with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the e cacy...

A pair of Wolfe type non-differentiable second order symmetric primal and dual problems in mathematical programming is formulated. The weak and strong duality theorems are then established under second order F-convexity assumptions. Symmetric minimax mixed integer primal and dual problems are also investigated. 2002 Elsevier Science B.V. All rights reserved.

• The dual problem is always convex no matter if the primal problem is convex, i.e., g is always concave. • The primal and dual optimal values, f∗ and g∗, always satisfy weak duality: f∗ ≥ g∗. • Slater’s condition: for convex primal, if there is an x such that h1(x) < 0, · · · , hm(x) < 0 and l1(x) = 0, · · · , lr(x) = 0 (12.5) then strong duality holds: f∗ = g∗. Note that the condition can be ...

In a recent paper [8], Chan and Sun reported for semidefinite programming (SDP) that the primal/dual constraint nondegeneracy is equivalent to the dual/primal strong second order sufficient condition (SSOSC). This result is responsible for a number of important results in stability analysis of SDP. In this paper, we study duality of this type in nonlinear semidefinite programming (NSDP). We int...

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model...

The Abadie CQ (ACQ) for convex inequality systems is a fundamental notion in optimization and approximation theory. In terms of the contingent cone and tangent derivative, we extend the Abadie CQ to more general convex multifunction cases and introduce the strong ACQ for both multifunctions and inequality systems. Some seemly unrelated notions are unified by the new ACQ and strong ACQ. Relation...

Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are well-known and form the basis of the highly successful branch-and-cut method. It is rather less well-known that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutti...

The Lagrange dual function is: g(u, v) = min x L(x, u, v) The corresponding dual problem is: maxu,v g(u, v) subject to u ≥ 0 The Lagrange dual function can be viewd as a pointwise maximization of some affine functions so it is always concave. The dual problem is always convex even if the primal problem is not convex. For any primal problem and dual problem, the weak duality always holds: f∗ ≥ g...

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadr...