Global optimization for the multilevel European gas market system with nonlinear flow models on trees

The European gas market is implemented as an entry-exit system, which aims to decouple transport and trading of gas. It has been modeled in the literature as a multilevel problem, which contains a nonlinear flow model of gas physics. Besides the multilevel structure and the nonlinear flow model, the computation of so-called technical capacities is another major challenge. These lead to nonlinear adjustable robust constraints that are computationally intractable in general. We provide techniques to equivalently reformulate these nonlinear adjustable constraints as finitely many convex constraints including integer variables in the case that the underlying network is tree-shaped. We further derive additional combinatorial constraints that significantly speed up the solution process. Using our results, we can recast the multilevel model as a single-level nonconvex mixed-integer nonlinear problem, which we then solve on a real-world network, namely the Greek gas network, to global optimality. Overall, this is the first time that the considered multilevel entry-exit system can be solved for a real-world sized network and a nonlinear flow model.

A Variable Sample-Size Stochastic Quasi-Newton Method for Smooth and Nonsmooth Stochastic Convex Optimization

Classical theory for quasi-Newton schemes has focused on smooth, deterministic, unconstrained optimization, whereas recent forays into stochastic convex optimization have largely resided in smooth, unconstrained, and strongly convex regimes. Naturally, there is a compelling need to address nonsmoothness, the lack of strong convexity, and the presence of constraints. Accordingly, this paper presents a quasi-Newton framework that can process merely convex and possibly nonsmooth (but smoothable) stochastic convex problems. We propose a framework that combines iterative smoothing and regularization with a variance-reduced scheme reliant on using an increasing sample size of gradients. We make the following contributions. (i) We develop a regularized and smoothed variable sample-size BFGS update (rsL-BFGS) that generates a sequence of Hessian approximations and can accommodate nonsmooth convex objectives by utilizing iterative regularization and smoothing. (ii) In strongly convex regimes with state-dependent noise, the proposed variable sample-size stochastic quasi-Newton (VS-SQN) scheme admits a nonasymptotic linear rate of convergence, whereas the oracle complexity of computing an [Formula: see text]-solution is [Formula: see text], where [Formula: see text] denotes the condition number and [Formula: see text]. In nonsmooth (but smoothable) regimes, using Moreau smoothing retains the linear convergence rate for the resulting smoothed VS-SQN (or sVS-SQN) scheme. Notably, the nonsmooth regime allows for accommodating convex constraints. To contend with the possible unavailability of Lipschitzian and strong convexity parameters, we also provide sublinear rates for diminishing step-length variants that do not rely on the knowledge of such parameters. (iii) In merely convex but smooth settings, the regularized VS-SQN scheme rVS-SQN displays a rate of [Formula: see text] with an oracle complexity of [Formula: see text]. When the smoothness requirements are weakened, the rate for the regularized and smoothed VS-SQN scheme rsVS-SQN worsens to [Formula: see text]. Such statements allow for a state-dependent noise assumption under a quadratic growth property on the objective. To the best of our knowledge, the rate results are among the first available rates for QN methods in nonsmooth regimes. Preliminary numerical evidence suggests that the schemes compare well with accelerated gradient counterparts on selected problems in stochastic optimization and machine learning with significant benefits in ill-conditioned regimes.

AN EFFICIENT HYBRID DERIVATIVE-FREE PROJECTION ALGORITHM FOR CONSTRAINT NONLINEAR EQUATIONS

In this paper, by combining the Solodov and Svaiter projection technique with the conjugate gradient method for unconstrained optimization proposed by Mohamed et al. (2020), we develop a derivative-free conjugate gradient method to solve nonlinear equations with convex constraints. The proposed method involves a spectral parameter which satisfies the sufficient descent condition. The global convergence is proved under the assumption that the underlying mapping is Lipschitz continuous and satisfies a weaker monotonicity condition. Numerical experiment shows that the proposed method is efficient.

Scheduling of a Microgrid with High Penetration of Electric Vehicles Considering Congestion and Operations Costs

This paper reviews the impact that can be presented by the immersion of generation sources and electric vehicles into the distribution network, with a technical, operational and commercial approach, given by the energy transactions between customer and operator. This requires a mathematical arrangement to identify the balance between congestion and the operating cost of a microgrid when the operation scheduling of the system a day ahead of horizon time it is required. Thus, this research is directed to the solution, using heuristic algorithms, since they allow the non-convex constraints of the proposed mathematical problem. The optimization algorithm proposed for the analysis is given by the Multi-Object Particle Swarm Optimization (MOPSO) method, which provides a set of solutions that are known as Optimal Pareto. This algorithm is presented in an IEEE 141-bus system, which consists of a radial distribution network that considers 141 buses used by Matpower; this system was modified and included a series of renewable generation injections, systems that coordinate electric vehicles and battery storage, and the slack node was maintained and assumed to have (traditional generation). In the end it can be shown that the algorithm can provide solutions for network operation planning, test system robustness and verify some contingencies comparatively, always optimizing the balance between congestion and cost.

A parameter-free unconstrained reformulation for nonsmooth problems with convex constraints

In the present paper we propose to rewrite a nonsmooth problem subjected to convex constraints as an unconstrained problem. We show that this novel formulation shares the same global and local minima with the original constrained problem. Moreover, the reformulation can be solved with standard nonsmooth optimization methods if we are able to make projections onto the feasible sets. Numerical evidence shows that the proposed formulation compares favorably against state-of-art approaches. Code can be found at https://github.com/jth3galv/dfppm.

An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications

This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced.