The main objective of the paper is to analyze how policymakers influence the random oligopolistic market equilibrium problem. To this purpose, random optimal control equilibrium conditions are introduced. Since the random optimal regulatory tax is characterized by a stochastic inverse variational inequality, existence and well-posedness results on such an inequality are proved. At last a numerical example is discussed.

We study large-scale portfolio optimization problems in which the aim is to maximize a multi-moment performance measure extending the Sharpe ratio. More specifically, we consider the adjusted for skewness Sharpe ratio, which incorporates the third moment of the returns distribution, and the adjusted for skewness and kurtosis Sharpe ratio, which exploits in addition the fourth moment. Further, we account for two types of real-world trading constraints. On the one hand, we impose stock market restrictions through cardinality, buy-in thresholds, and budget constraints. On the other hand, a turnover threshold restricts the total allowed amount of trades in the rebalancing phases. To deal with these asset allocation models, we embed a novel hybrid constraint-handling procedure into an improved dynamic level-based learning swarm optimizer. A repair operator maps candidate solutions onto the set characterized by the first type of constraints. Then, an adaptive ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-exact penalty function manages turnover violations. The focus of the paper is to highlight the importance of including higher-order moments in the performance measures for long-run investments, in particular when the market is turbulent. We carry out empirical tests on two worldwide sets of assets to illustrate the scalability and effectiveness of the proposed strategies, and to evaluate the performance of our investments compared to the strategy maximizing the Sharpe ratio.

In this paper we provide an efficient methodology to compute the credit value adjustment of a European contingent claim subject to some default event concerning the issuer solvability, when the underlying and the default event are correlated. In particular, in a Black and Scholes market/CIR intensity-default model, we consider a second order expansion around the origin of a vulnerable call option with respect to a correlation parameter rho, which may be used to describe the wrong way risk of the contract, measuring the dependence between the underlying asset price and the option’s issuer default intensity. Numerical implementations of this approach are compared with the benchmark Monte Carlo simulations.

We study relations between the solution sets of Variational Inequalities, Minty Variational Inequalities, Natural Map problems and Nash Equilibrium Problems. Moreover, motivated by the inherent relevance of inexactness both in modeling non-cooperative games and in algorithms for variational inequalities, we consider inexact versions of such problems and we establish relations to quantify how inexactness propagates from one problem to the other.

Given two sets of objects, the integer assignment problem consists of assigning objects of one set to objects in the other set. Traditionally, the goal of this problem is to find an assignment that minimizes or maximizes a measure of efficiency, such as maximization of utility or minimization of cost. Lately, the interest in incorporating a measure of fairness in addition to efficiency has gained importance. This paper studies how to incorporate these two criteria in an integer assignment, using the Jain’s index as a measure of fairness. The original formulation of the assignment problem with this index involves a non-concave function, which renders a non-linear non-convex problem, usually hard to solve. To this aim, we develop two reformulations, where one is based on a quadratic objective function and the other one is based on integer second-order cone programming. We explore the performance of these reformulations in instances of real-world data derived from an application of assigning personnel to projects, and also in instances of randomly generated data. In terms of solution quality, all formulations prove to be effective in finding solutions capturing both efficiency and fairness criteria, with some slight differences depending on the type of instance. In terms of solving time, however, the performances of the formulations differ considerably. In particular, the integer quadratic approach proves to be much faster in finding optimal solutions.

We formulate and compare optimization models of investment in renewable generation using a suite of social planning models that compute optimal generation capacity investments for a hydro-dominated electricity system where inflow uncertainty results in a risk of energy shortage. The models optimize the expected cost of capacity expansion and operation allowing for investments in hydro, geothermal, solar, wind, and thermal plant, as well as battery storage for smoothing load profiles. A novel feature is the integration of uncertain seasonal hydroelectric energy supply and short-term variability in renewable supply in a two-stage stochastic programming framework. The models are applied to data from the New Zealand electricity system and used to estimate the costs of moving to a 100% renewable electricity system by 2035. We also explore the outcomes obtained when applying different forms of CO2 constraint that limit respectively non-renewable capacity, non-renewable generation, and CO2 emissions on average, almost surely, or in a chance-constrained setting, and show how our models can be used to investigate the merits of a proposed pumped-hydro scheme in New Zealand’s South Island.

We present two easy-to-implement gradient-free/zeroth-order methods to optimize a stochastic non-smooth function accessible only via a black-box. The methods are built upon efficient first-order methods in the heavy-tailed case, i.e., when the gradient noise has infinite variance but bounded (1 + κ)-th moment for some κ ∈ (0, 1]. The first algorithm is based on the stochastic mirror descent with a particular class of uniformly convex mirror maps which is robust to heavy-tailed noise. The second algorithm is based on the stochastic mirror descent and gradient clipping technique. Additionally, for the objective functions satisfying the r-growth condition, faster algorithms are proposed based on these methods and the restart technique.