Dissecting the duality gap: the supporting hyperplane interpretation revisited

We revisit the classic supporting hyperplane illustration of the duality gap for non-convex optimization problems. It is refined by dissecting the duality gap into two terms: the first measures the degree of near-optimality in a Lagrangian relaxation, while the second measures the degree of near-complementarity in the Lagrangian relaxed constraints. We also give an example of how this dissection may be exploited in the design of a solution approach within discrete optimization.

Finding Equilibria in the Traffic Assignment Problem with Primal-Dual Gradient Methods for Stable Dynamics Model and Beckmann Model

In this paper, we consider the application of several gradient methods to the traffic assignment problem: we search equilibria in the stable dynamics model (Nesterov and De Palma, 2003) and the Beckmann model. Unlike the celebrated Frank–Wolfe algorithm widely used for the Beckmann model, these gradients methods solve the dual problem and then reconstruct a solution to the primal one. We deal with the universal gradient method, the universal method of similar triangles, and the method of weighted dual averages and estimate their complexity for the problem. Due to the primal-dual nature of these methods, we use a duality gap in a stopping criterion. In particular, we present a novel way to reconstruct admissible flows in the stable dynamics model, which provides us with a computable duality gap.

Improved Lagrangian relaxation generation decision-support in presence of electric vehicles

<p><span>Decision making strategies for resources available in macro/micro scales have long been a critical argument. Among existing methods to address such a mixed-binary optimization model, Lagrangian relaxation (LR) found universal acceptance by many utilities, offering a fast and accurate answer. This paper aims at retrofitting the solution way of LR algorithm by dint of meta-heuristic cuckoo search algorithm (CSA). When integrating CSA into LR mechanism, a tighter duality gap is catered, representing more accurate feasible solution. The key performance of CSA exhibits a head start over other classical methods such as gradient search (GS) and Newton Raphson (NR) when dealt with the relative duality gap closure in LR procedure. Further, electric vehicles (EV) with its associated hard constraints are encompassed into model to imperiling the proposed CSA-LR if encountered with nonlinear fluctuation of duality gap. Simulation results show that the proposed CSA-LR model outperforms the solution quality with/without EV as compared with conventional NR-LR method.</span></p>

A new proof of the dual optimization problem and its application to the optimal material distribution of SiC/graphene composite

This paper presents a simple and direct proof of the dual optimization problem. The stationary conditions of the original and the dual problems are exactly equivalent, and the duality gap can be completely eliminated in the dual problem, where the maximal or minimal value is solved together with the stationary conditions of the dual problem and the original constraints. As an illustration, optimization of SiC/graphene composite is addressed with an objective of maximizing certain material properties under the constraint of a given strength.