Modeling of Competition and Collaboration Networks under Uncertainty: Stochastic Programs with Resource and Bilevel

We analyze stochastic programming problems with recourse characterized by a bilevel structure. Part of the uncertainty in such problems is due to actions of other actors such that the considered decision maker needs to develop a model to estimate their response to his decisions. Often, the resulting model exhibits connecting constraints in the leaders (upper-level) subproblem. It is shown that this problem can be formulated as a new class of stochastic programming problems with equilibrium constraints (SMPEC). Sufficient optimality conditions are stated. A solution algorithm utilizing a stochastic quasi-gradient method is proposed, and its applicability extensively explained by practical numerical examples

Stochastic gradient approach for energy and supply optimisation in water systems management

Under conditions of water scarcity, energy saving in operation of water pumping plants and the minimisation of water deficit for users and activities are frequently contrasting requirements, which should be considered when optimising large-scale multi-reservoirs and multi-users water supply systems. Undoubtedly, a high uncertainty level in predicted water resources due to hydrologic input variability and water demand behaviour characterizes this problem. The aim of this paper is to provide an efficient decision support system considering emergency water pumping plants activation schedules. The obtained results should allow the water system’s authority to adopt a robust decision policy, minimising the risk of harmful future decisions concerning the water resource management. The model has been here developed to manage this problem, in order to reduce the damages due to shortage of water and the energy-cost requirements of pumping plants. Particularly, in optimisation, we look for optimal rules considering both historical and generated synthetic scenarios of hydrologic inputs to reservoirs. Hence, using synthetic series, we can analyse climate change impacts and optimise the activation rules considering future hydrologic occurrences. A simulation model has been coupled with an optimization module using the stochastic gradient method to get robust pumping activation thresholds. This method allows to solve complex problems, solving efficiently large size real cases due to high number of data and variables. Thresholds values are identified in terms of critical storage levels in supply-reservoirs. Application of the modelling approach has been developed on a real case study in a water-shortage prone area in south-Sardinia (Italy), characterized by Mediterranean climate and high annual variability in hydrological input to reservoirs. By applying the combined simulation procedure, a robust decision strategy in pumping activation was obtained. Developing the stochastic gradient model, a main programming supports has been built by MATLAB efficiently interfaced with CPLEX for optimisation and Excel for inputs and results representation

Stochastic gradient methods for energy saving and a correct management in complex water supply systems

The management optimization of complex multi-source and multi-demand water resource systems under a high uncertainty level has been a subject of interest in the research literature (Labadie, 2004; Cunha & Sousa, 2010; Yuan et al., 2016). In this context, energy saving in operation of water pumping plants and reduction of water deficit for users and activities are frequently conflicting issues. Dealing with these problems, the definition of optimal activation rules for emergency activation of pumping stations are a relevant topic recently treated in Lerma et al. (2015) and Napolitano et al. (2016). In this study we want to define a trade-off between costs and risks considering the minimization of water shortage damages and the pumping operative costs, under different hydrological scenarios occurrences possibilities. Consequently, optimization results should provide the water system Authorities with a robust information about the optimal activation rules considering a large set of generated scenarios of hydrologic inputs to reservoirs. Using synthetic series it is possible to take into account the climate change impacts and balance the rules while also considering future behavior under the risk of the occurrence of shortages and the cost of early warning procedures to avoid water scarcity, mainly related to activation of emergency water transfers. Thereafter, this problem has been faced considering an efficient optimization tool based on the Stochastic Gradient method (SQG), see Ermoliev & Wets (1988) and Gaivoronski (2005). Testing the effectiveness of this proposal, an application of the modelling approach has been developed in a water shortage prone area in South-Sardinia (Italy)

Nonstationary Optimization Approach for Finding Universal Portfolios

The definition of universal portfolio was introduced in the nancial literature in order to describe the class of portfolios which are constructed directly from the available observations of the stocks behavior without any assumptions about their statistical properties. Cover has shown that one can construct such portfolio using only observations of the past stock prices which generates the same asymptotic wealth growth as the best constant rebalanced portfolio which is constructed with the full knowledge of the future stock market behavior. In this paper we construct universal portfolios using totally different set of ideas drawn from nonstationary stochastic optimization. Also our portfolios yield the same asymptotic growth of wealth as the best constant rebalanced portfolio constructed with the perfect knowledge of the future, but they are less demanding computationally. Besides theoretical study, we present computational evidence using data from New York Stock Exchange which shows, among other things, superior performance of portfolios which explicitly take into account possible nonstationary market behavior

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page

Optimization of complex simulation models with stochastic gradient methods

We describe the structure of stochastic optimization solver SQG (Stochastic QuasiGradient), which implements stochastic gradient methods for optimization of complex stochastic simulation models. The solver finds the equilibrium solution when the simulation model describes the system with several actors. The solver is parallelizable and it performs several simulation threads in parallel. It is capable of solving stochastic optimization problems, finding stochastic Nash equilibria, stochastic bilevel problems where each level may require the solution of stochastic optimization problem or finding Nash equilibrium. We provide several complex examples with applications to water resources management, energy markets, pricing of services on social networks

Linearization methods for optimization of functionals which depend on probability measures

The main purpose of this paper is to discuss numerical optimization procedures for problems in which both the objective function and the constraints depend on distribution functions. The objective function and constraints are assumed to be nonlinear and to have directional derivatives. The proposed algorithm is based on duality relations between the linearized problem and some special finite-dimensional minimax problem and is of the feasible-direction type. The resulting minimax problem is solved using the cutting-place technique

Stochastic optimization techniques for finding optimal submeasures

In this paper, the author looks at some quite general optimization problems on the space of probabilistic measures. These problems originated in mathematical statistics but have applications in several other areas of mathematical analysis. The author extends previous work by considering a more general form of the constraints, and develops numerical methods (based on stochastic quasigradient techniques) and some duality relations for problems of this type. This paper is a contribution to research on stochastic optimization currently underway within the Adaptation and Optimization Project