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We study nested variational inequalities, which are variational inequalities whose feasible set is the solution set of another variational inequality. We present a projected averaging Tikhonov algorithm requiring the weakest conditions in the literature to guarantee the convergence to solutions of the nested variational inequality. Specifically, we only need monotonicity of the upper- and the lower-level variational inequalities. Also, we provide the first complexity analysis for nested variational inequalities considering optimality of both the upper- and lower-level.

We address the minimization of an objective function over the solution set of a (non-parametric) lower-level variational inequality. This problem is a special instance of semi-infinite programs and encompasses, as particular cases, simple (smooth) bilevel and equilibrium selection problems. We resort to a suitable approximated version of the hierarchical problem. We show that this, on the one hand, does not perturb the original (exact) program ‘too much’, on the other hand, allows one to rely on some suitable exact penalty approaches whose convergence properties are established.

We consider potential games with mixed-integer variables, for which we propose two distributed, proximal-like equilibrium seeking algorithms. Specifically, we focus on two scenarios: i) the underlying game is generalized ordinal and the agents update through iterations by choosing an exact optimal strategy; ii) the game admits an exact potential and the agents adopt approximated optimal responses. By exploiting the properties of integer-compatible regularization functions used as penalty terms, we show that both algorithms converge to either an exact or an $\epsilon$-approximate equilibrium. We corroborate our findings on a numerical instance of a Cournot oligopoly model.

In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability of our scheme compared with some related methods in the literature.

Following the research strands of enhanced index tracking and of portfolio performance measures optimization, we propose to choose, among the feasible asset portfolios of a given market, the one that maximizes the geometric mean of the differences between its risk and gain and those of a suitable reference benchmark, such as the market index. This approach, which has a peculiar geometric interpretation and enjoys remarkable features, provides the efficient portfolio that dominates the largest amount of portfolios dominating the reference benchmark index. Preliminary empirical results highlight good out-of-sample performances of our approach compared with those of the market index.

Distributed and parallel algorithms have been frequently investigated in the recent years, in particular in applications like machine learning. Nonetheless, only a small subclass of the optimization algorithms in the literature can be easily distributed, for the presence, e.g., of coupling constraints that make all the variables dependent from each other with respect to the feasible set. Augmented Lagrangian methods are among the most used techniques to get rid of the coupling constraints issue, namely by moving such constraints to the objective function in a structured, well-studied manner. Unfortunately, standard augmented Lagrangian methods need the solution of a nested problem by needing to (at least inexactly) solve a subproblem at each iteration, therefore leading to potential inefficiency of the algorithm. To fill this gap, we propose an augmented Lagrangian method to solve convex problems with linear coupling constraints that can be distributed and requires a single gradient projection step at every iteration. We give a formal convergence proof to at least $$\varepsilon $$-approximate solutions of the problem and a detailed analysis of how the parameters of the algorithm influence the value of the approximating parameter $$\varepsilon $$. Furthermore, we introduce a distributed version of the algorithm allowing to partition the data and perform the distribution of the computation in a parallel fashion.

We present a mixed integer nonlinear mathematical programming model, covering a broad range of operations research (OR)-related topics. The case is designed to allow students to use knowledge acqui...

Urged by the outbreak of the COVID-19 in Italy, this study aims at helping to tackle the spread of the disease by resorting to operations research techniques. In particular, we propose a mathematical program to model the problem of establishing how many diagnostic tests the Italian regions must perform in order to maximize the overall disease detection capability. An important feature of our approach is its simplicity: data we resort to are easy to obtain and one can employ standard optimization tools to address the problem. The results we obtain when applying our method to the Italian case seem promising.

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branc...

We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explicitly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our ``uneven'' horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our ``uneven'' horizontal model, in some sense, lies between the vertical bilevel model and a ``pure'' horizontal game.

We define the concept of reproducible map and show that, whenever the constraint map defining the quasivariational inequality (QVI) is reproducible then one can characterize the whole solution set of the QVI as a union of solution sets of some variational inequalities (VI). By exploiting this property, we give sufficient conditions to compute any solution of a generalized Nash equilibrium problem (GNEP) by solving a suitable VI. Finally, we define the class of pseudo-Nash equilibrium problems, which are (not necessarily convex) GNEPs whose solutions can be computed by solving suitable Nash equilibrium problems.

Relying on multiobjective programming techniques, we have developed an optimization software to improve the services provided by the social co-operative OMNIA. OMNIA’s mission is to supply home-care assistance to children and people with disabilities. Our method is intended to optimize the social workers’ shift planning aiming at, on the one hand, maximizing the overall quality of the social care services, on the other hand, minimizing costs associated with OMNIA’s activity. In particular, our software provides Pareto optima of the resulting (difficult) bi-objective model by resorting to both a standard a priori weighted-sum approach and a new MINLP no-preference (hypervolume maximization-type) method. The product of this research is successfully employed by OMNIA.

We propose a new formulation of the Karush---Kunt---Tucker conditions of a particular class of quasi-variational inequalities. In order to reformulate the problem we use the Fisher---Burmeister complementarity function and canonical duality theory. We establish the conditions for a critical point of the new formulation to be a solution of the original quasi-variational inequality showing the potentiality of such approach in solving this class of problems. We test the obtained theoretical results with a simple heuristic that is demonstrated on several problems coming from the academy and various engineering applications.

We propose to solve a general quasi-variational inequality by using its Karush–Kuhn–Tucker conditions. To this end we use a globally convergent algorithm based on a potential reduction approach. We establish global convergence results for many interesting instances of quasi-variational inequalities, vastly broadening the class of problems that can be solved with theoretical guarantees. Our numerical testings are very promising and show the practical viability of the approach.

We present an adaptive system for the automatic assessment of both physical and anthropic fire impact factors on periurban forestries. The aim is to provide an integrated methodology exploiting a complex data structure built upon a multi resolution grid gathering historical land exploitation and meteorological data, records of human habits together with suitably segmented and interpreted high resolution X-SAR images, and several other information sources. The contribution of the model and its novelty rely mainly on the definition of a learning schema lifting different factors and aspects of fire causes, including physical, social and behavioural ones, to the design of a fire susceptibility map, of a specific urban forestry. The outcome is an integrated geospatial database providing an infrastructure that merges cartography, heterogeneous data and complex analysis, in so establishing a digital environment where users and tools are interactively connected in an efficient and flexible way.

We consider the problem of optimization in policy space for reinforcement learning. While a plethora of methods have been applied to this problem, only a narrow category of them proved feasible in robotics. We consider the peculiar characteristics of reinforcement learning in robotics, and devise a combination of two algorithms from the literature of derivative-free optimization. The proposed combination is well suited for robotics, as it involves both off-line learning in simulation and on-line learning in the real environment. We demonstrate our approach on a real-world task, where an Autonomous Underwater Vehicle has to survey a target area under potentially unknown environment conditions. We start from a given controller, which can perform the task under foreseeable conditions, and make it adaptive to the actual environment.

When dealing with big data problems it is crucial to design methods able to decompose the original problem into smaller and more manageable pieces. Parallel methods lead to a solution by concurrently working on different pieces that are distributed among available agents, so that exploiting the computational power of multi-core processors and therefore efficiently solving the problem. Beyond gradient-type methods, that can of course be easily parallelized but suffer from practical drawbacks, recently a convergent decomposition framework for the parallel optimization of (possibly non-convex) big data problems was proposed. Such framework is very flexible and includes both fully parallel and fully sequential schemes, as well as virtually all possibilities in between. We illustrate the versatility of this parallel decomposition framework by specializing it to different well-studied big data problems like LASSO, logistic regression and support vector machines training. We give implementation guidelines and numerical results showing that proposed parallel algorithms work very well in practice.

Inspired by the optimal value approach, we propose a new reformulation of the optimistic Bilevel programming Problem (BP) as a suitable Generalized Nash Equilibrium Problem (GNEP). We provide a complete analysis of the relationship between the original hierarchical BP and the corresponding "more democratic" GNEP. Moreover, we investigate solvability and convexity issues of our reformulation. Finally, relying on the vast literature on solution methods for GNEPs, we devise a new effective algorithmic framework for the solution of signicant classes of BPs.

We propose a decomposition framework for the parallel optimization of the sum of a differentiable function and a (block) separable nonsmooth, convex one. The latter term is typically used to enforce structure in the solution as, for example, in Lasso problems. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss-Seidel (Southwell-type) ones, as well as virtually all possibilities in between (e.g., gradient- or Newton-type methods) with only a subset of variables updated at each iteration. Our theoretical convergence results improve on existing ones, and numerical results show that the new method compares favorably to existing algorithms., Comment: submitted to IEEE ICASSP 2014

In this paper Synthetic Aperture Radar (SAR) images in X-band were analyzed in order to infer ground properties from data. The aim was to classify different zones in peri-urban forestries integrating information from different sources. In particular the X band is sensitive to the moisture content of the ground that can be therefore put into relation with the gray level of the image; moreover, the gray level is related to the smoothness and roughness of the ground. An integration of image segmentation and machine learning methods is studied to classify different zones of peri-urban forestries, such as trees canopies, lawns, water pounds, roads, etc., directly from the gray level signal properties. As case study the X-SAR data of a forest near Rome, the Castel Fusano area, are analyzed.

Our work aims to the implementation and numerical evaluation of multithread methods within the context of global optimization algorithms for complex problems. In particular we develop asynchronous parallel derivative-free algorithms using both MPI and OpenMP paradigms. Evaluation and calibration of these algorithms is performed by using the HPC cluster Matrix of the "Consorzio interuniversitario per le Applicazioni di Supercalcolo Per Universit`a e Ricerca" (CASPUR). Numerical results are presented for applications in robust design optimization of high speed catamarans

The following article has been included in a multiple retraction: Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract Global Optimization Solutions to a Class of Non-convex Quadratic Minimization Problems with Quadratic Constraints; Yubo Yuan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591086.abstract On Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao. http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstract The Editor-in-Chief and SAGE strive to uphold the very highest standards of publication ethics and are committed to supporting the high standards of integrity of Mathematics & Mechanics of Solids. Authors, reviewers, editors and interested readers are encouraged to consult SAGE’s ethics statements and the Committee on Publication Ethics (COPE) website for guidelines on publication ethics.

In this paper we present an integrated approach to COSMO-SkyMed image analysis and classification exploiting integration of different data of the regions of interest, namely urban forestry areas, wide urban parks. The aim is to provide a methodology for exploiting complex data structures built upon multi resolution grids gathering together with optical and X-SAR images, also historical land exploitation and meteorological data, records of human habits, and several other information sources. Although these data are specifically gathered to built a fire susceptibility map, the method is quite general. Indeed, the contribution of the model and its novelty relies manly on the definition of a learning schema lifting different factors and aspects of the event to be identified (here fire causes), including physical, social and behavioral ones, to the design of a fire susceptibility map, of a specific urban forestry. The outcome is an integrated geospatial database providing an infrastructure that merges cartography, heterogeneous data and complex analysis, in so establishing a digital environment where users and tools are interactively connected in an efficient and flexible way.