We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

This paper presents an optimal power management mechanism for grid connected photovoltaic (PV) systems with storage. The objective is to help intensive penetration of PV production into the grid by proposing peak shaving service at the lowest cost. The structure of a power supervisor based on an optimal predictive power scheduling algorithm is proposed. Optimization is performed using Dynamic Programming and is compared with a simple ruled-based management. The particularity of this study remains first in the consideration of batteries ageing into the optimization process and second in the “day-ahead” approach of power management. Simulations and real conditions application are carried out over one exemplary day. In simulation, it points out that peak shaving is realized with the minimal cost, but especially that power fluctuations on the grid are reduced which matches with the initial objective of helping PV penetration into the grid. In real conditions, efficiency of the predictive schedule depends on accuracy of the forecasts, which leads to future works about optimal reactive power management.

Optimal Power Flow Management for Grid Connected PV Systems With Batteries

In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Flows in Networks

Deep learning (DL) techniques have obtained remarkable achievements on various tasks, such as image recognition, object detection, and language modeling. However, building a high-quality DL system for a specific task highly relies on human expertise, hindering its wide application. Meanwhile, automated machine learning (AutoML) is a promising solution for building a DL system without human assistance and is being extensively studied. This paper presents a comprehensive and up-to-date review of the state-of-the-art (SOTA) in AutoML. According to the DL pipeline, we introduce AutoML methods – covering data preparation, feature engineering, hyperparameter optimization, and neural architecture search (NAS) – with a particular focus on NAS, as it is currently a hot sub-topic of AutoML. We summarize the representative NAS algorithms’ performance on the CIFAR-10 and ImageNet datasets and further discuss the following subjects of NAS methods: one/two-stage NAS, one-shot NAS, joint hyperparameter and architecture optimization, and resource-aware NAS. Finally, we discuss some open problems related to the existing AutoML methods for future research.

AutoML: A survey of the state-of-the-art

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

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Mixed-integer nonlinear optimization

The paper deals with a unit commitment problem of a generation company whose aim is to find the optimal scheduling of a multiunit pump-storage hydro power station, for a short term period in which the electricity prices are forecasted. The problem has a mixed-integer nonlinear structure, which makes very hard to handle the corresponding mathematical models. However, modern mixed-integer linear programming (MILP) software tools have reached a high efficiency, both in terms of solution accuracy and computing time. Hence we introduce MILP models of increasing complexity, which allow to accurately represent most of the hydroelectric system characteristics, and turn out to be computationally solvable. In particular we present a model that takes into account the head effects on power production through an enhanced linearization technique, and turns out to be more general and efficient than those available in the literature. The practical behavior of the models is analyzed through computational experiments on real-world data.

/pdf/an-milp-approach-for-short-term-hydro-scheduling-and-unit-z08hlc7hgy.pdf

An MILP Approach for Short-Term Hydro Scheduling and Unit Commitment With Head-Dependent Reservoir

We propose a practical solution method for real-world instances of a water-network optimization problem with fixed topology using a nonconvex continuous NLP (NonLinear Programming) relaxation and a MINLP (Mixed Integer NonLinear Programming) search. Our approach employs a relatively simple and accurate model that pays some attention to the requirements of the solvers that we employ. Our view is that in doing so, with the goal of calculating only good feasible solutions, complicated algorithmics can be confined to the MINLP solver. We report successful computational experience using available open-source MINLP software on problems from the literature and on difficult real-world instances. An important contribution of this paper is that the solutions obtained, besides being low cost, are immediately usable in practice because they are characterized by an allocation of diameters to pipes that leads to a correct hydraulic operation of the network. This is not the case for most of the other methods presented in the literature.

Claudia D’Ambrosio

Papers

An MILP Approach for Short-Term Hydro Scheduling and Unit Commitment With Head-Dependent Reservoir

On the optimal design of water distribution networks: a practical MINLP approach

Piecewise linear approximation of functions of two variables in MILP models

Surrogate‐based methods for black‐box optimization

Mathematical Programming techniques in Water Network Optimization