Editorial for the Special Issue on “Recent Advances in Nonsmooth Optimization and Analysis”

In recent years, nonsmooth optimization and analysis have seen remarkable advancements, significantly impacting various scientific and engineering disciplines. Nonsmooth optimization deals with optimization problems, where the objective function or the constraints are not differentiable. Such problems arise naturally in numerous real-world applications, including machine learning, game theory, logistics, networks, signal processing, and control theory. The ability to handle nonsmoothness effectively opens new avenues for solving complex problems that were previously intractable. This Special Issue of Algorithms, titled “Recent Advances in Nonsmooth Optimization and Analysis”, aimed to contribute to the research into and development of these fields. This editorial provides an overview of the contributions, and the practical applications presented in them.

The survey titled “A state of the art review of systems of linear inequalities and related observability problems” reviews the state of the art on the use, disclosure, and propagation of systems of linear inequalities in real life, teaching, and research. These are connected to compatibility and observability problems. Illustrating applications in various areas are presented, highlighting the advantages of solving systems of inequalities which, despite an apparent simplicity, turn out to be efficient tools for addressing a large variety of (more) complicated problems. In particular, their practical relevance in artificial intelligence and automatic learning is stressed, strengthening the idea that effective and efficient methods for solving systems of inequalities are widely relevant beyond their immediate usage.

Another theoretical contribution is the article titled “Lipschitz continuity results for a class of parametric variational inequalities and applications to network games”, where the Lipschitz continuity of a class of parametric variational inequalities is addressed. New estimates for the Lipschitz constant of the solutions of finitely dimensional variational inequalities, where both the operator and the constraint set can depend on a parameter, are provided, in particular, improving their previous counterparts from the literature. An algorithm is then proposed to solve the problem of computing the mean value of such a solution with respect to the involved parameter. Applications to Nash equilibrium problems, including generalized Nash equilibrium problems on networks, illustrate potential practical outcomes of these theoretical advances.

The article titled “A linearly involved generalized Moreau enhancement of ${\ell }_{2,1}$-norm with application to weighted group sparse classification” proposes a new group-sparsity-inducing nonconvex regularizer for approximating the ${\ell }_{2,0}$ pseudo-norm. The new regularizer can be seen as a linearly involved generalized Moreau enhancement of the ${\ell }_{2,1}$ norm. The proposed model can handle general group configurations, such as weighted group sparse problems, and splitting proximal point type algorithms can be employed for solving it. Various potential applications are discussed, and computational results on (weighted) group sparse classification problems show that the proposed classifier improves the performance of convex ${\ell }_{2,1}$ regularizer-based methods, in particular for unbalanced training data sets.

As already mentioned above, the algorithmic side of nonsmooth optimization is also present in this Special Issue, in the article titled “On the adaptive penalty parameter selection in ADMM”, where the adaptive penalty parameter selection in the classical algorithm Alternate Direction Method of Multipliers (ADMM) is studied, motivated by earlier experiences showing that the choice of the parameter penalizing the constraint violation in the Augmented Lagrangian underlying ADMM affects the performance of the method. The authors investigate a recently proposed adaptive spectral strategy by employing different strategies based on Barzilai–Borwein-like stepsize rules. Computational experiments on real-life consensus logistic regression and portfolio optimization problems highlight the advances brought to the existing literature by the proposed approach.

A convex optimization approach is proposed in the article titled “A convex optimization algorithm for electricity pricing of charging stations” for solving the problem of electricity pricing for charging stations, motivated by the low performance of existing approaches for solving this multi-objective mixed integer nonlinear programming problem. The key step in this approach consists of transforming the initial problem into a convex optimization one via second-order conic relaxation and Karush–Kuhn–Tucker optimality conditions. The obtained optimization problem is solved many times in order to derive a good external approximation of the corresponding Pareto frontier. Experiments based on an IEEE 33-bus distribution network model show that the proposed method outperforms the currently widely-used heuristic ones.

An application in wireless multimedia sensors networks is discussed in the article titled “A cross-layer optimization QoS scheme in wireless multimedia sensor networks”. A joint flow control, routing, scheduling, and power control scheme, based on a Lyapunov optimization framework, is proposed to increase network lifetime and scheduling fairness, involving a differentiated queuing service for an adaptive distribution of transmission opportunities. As the power control problem turns out to be convex, two algorithms for solving it are discussed. Experimental results reveal that the new approach outperforms existing ones by achieving a better trade-off between QoS performances and network lifetime, in particular in the transmission of real-time services.

Last but not least, the article titled “Optimal maintenance schedule for a wind power turbine with aging components” proposes an optimal preventive maintenance algorithm for wind turbines, which has obvious direct applications in today’s economy. The method is based on a new notion of virtual maintenance that allows treating each replacement event as a renewal one, even when some components are not replaced by new ones, and exploits the renewal-reward theorem. The new algorithm is tested on a four-component model of a wind turbine, for which the optimal maintenance plans are determined under various initial conditions. The performed experiments show that the newly proposed approach lowers the maintenance costs by at about a third in comparison with baseline methods.

In conclusion, this Special Issue on recent advances in nonsmooth optimization and analysis presents a sample of contemporary research that underscores the vitality and importance of this field. The contributions from authors working in different countries across several continents not only address theoretical questions, but also demonstrate practical applications in fields lying in the focus of today’s society and economy (such as wind energy, electrical networks, and wireless multimedia sensors) that can drive progress in various industries. As we look to the future, continued research into nonsmooth optimization will undoubtedly yield even more profound insights and innovations.