(Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution

We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in case of linear random inequality systems. In the case of a constant coefficient matrix an upper estimate for even the smaller Mordukhovich subdifferential is proven

Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions

Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. In order to do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be successfully done by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz' code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, De'ak's sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used in order to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. Later, the result is extended to alternative distributions with an emphasis on the multivariate Student (or T-) distribution

Incremental bundle methods using upper models

We propose a family of proximal bundle methods for minimizing sum-structured convex nondifferentiable functions which require two slightly uncommon assumptions, that are satisfied in many relevant applications: Lipschitz continuity of the functions and oracles which also produce upper estimates on the function values. In exchange, the methods: i) use upper models of the functions that allow to estimate function values at points where the oracle has not been called; ii) provide the oracles with more information about when the function computation can be interrupted, possibly diminishing their cost; iii) allow to skip oracle calls entirely for some of the component functions, not only at "null steps" but also at "serious steps"; iv) provide explicit and reliable a-posteriori estimates of the quality of the obtained solutions; v) work with all possible combinations of different assumptions on how the oracles deal with not being able to compute the function with arbitrary accuracy. We also discuss the introduction of constraints (or, more generally, of easy components) and use of (partly) aggregated models

Incremental bundle methods using upper models

We propose a family of proximal bundle methods for minimizing sum-structured convex nondifferentiable functions which require two slightly uncommon assumptions, that are satisfied in many relevant applications: Lipschitz continuity of the functions and oracles which also produce upper estimates on the function values. In exchange, the methods: i) use upper models of the functions that allow to estimate function values at points where the oracle has not been called; ii) provide the oracles with more information about when the function computation can be interrupted, possibly diminishing their cost; iii) allow to skip oracle calls entirely for some of the component functions, not only at ``null steps'' but also at ``serious steps''; iv) provide explicit and reliable a-posteriori estimates of the quality of the obtained solutions; v) work with all possible combinations of different assumptions on the oracles. We also discuss introduction of constraints (or, more generally, of easy components) and use of (partly) aggregated models

Generalized gradients for probabilistic/robust (probust) constraints

Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data

Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

We consider a profit-maximizing model for pricing contracts as an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A classical approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop an efficient pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming. These results are illustrated by an application to the optimal pricing of electricity contracts on the French market.Comment: 37 pages, 9 figures; adding a section on the pricing of electricity contract

Ergodic control of a heterogeneous population and application to electricity pricing

We consider a control problem for a heterogeneous population composed of customers able to switch at any time between different contracts, depending not only on the tariff conditions but also on the characteristics of each individual. A provider aims to maximize an average gain per time unit, supposing that the population is of infinite size. This leads to an ergodic control problem for a "mean-field" MDP in which the state space is a product of simplices, and the population evolves according to a controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we show that the ergodic eigenproblem admits a solution. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. We illustrate this approach on examples from electricity pricing, and show in particular that the optimal policies may be cyclic-alternating between discount and profit taking stages