What is the optimal cutoff surface for ore bodies with more than one mineral?

In mine planning problems, cutoff grade optimization defines a threshold at every time period such that material above this value is processed, and the rest is considered waste. In orebodies with multiple minerals, which occur in practice, the natural extension is to consider a cutoff surface. We show that in two dimensions the optimal solution … Read more

Contextual Chance-Constrained Programming

Uncertainty in classical stochastic programming models is often described solely by independent random parameters, ignoring their dependence on multidimensional features. We describe a novel contextual chance-constrained programming formulation that incorporates features, and argue that solutions that do not take them into account may not be implementable. Our formulation cannot be solved exactly in most cases, … Read more

Risk-Averse Multistage Stochastic Programs with Expected Conditional Risk Measures

We study decomposition algorithms for risk-averse multistage stochastic programs with expected conditional risk measures (ECRMs). ECRMs are attractive because they are time-consistent, which means that a plan made today will not be changed in the future if the problem is re-solved given a realization of the random variables. We show that solving risk-averse problems based … Read more

Multistage stochastic programs with the entropic risk measure

Over the last two decades, coherent risk measures have been well studied as a principled, axiomatic way to measure the risk of a random variable. Because of this axiomatic approach, coherent risk measures have a number of attractive features for computation, and they have been integrated into a variety of stochastic programming algorithms, including stochastic … Read more

The risk-averse ultimate pit problem

In this work, we consider a risk-averse ultimate pit problem where the grade of the mineral is uncertain. We propose a two-stage formulation of the problem and discuss which properties are desirable for a risk measure in this context. We show that the only risk measure that satisfies these properties is the entropic. We propose … Read more

Partially observable multistage stochastic programming

We propose a class of partially observable multistage stochastic programs and describe an algorithm for solving this class of problems. We provide a Bayesian update of a belief-state vector, extend the stochastic programming formulation to incorporate the belief state, and characterize saddle-function properties of the corresponding cost-to-go function. Our algorithm is a derivative of the … Read more

Scenario Reduction for Risk-Averse Stochastic Programs

In this paper we discuss scenario reduction methods for risk-averse stochastic optimization problems. Scenario reduction techniques have received some attention in the literature and are used by practitioners, as such methods allow for an approximation of the random variables in the problem with a moderate number of scenarios, which in turn make the optimization problem … Read more

An algorithm for binary chance-constrained problems using IIS

We propose an algorithm based on infeasible irreducible subsystems (IIS) to solve general binary chance-constrained problems. By leverag- ing on the problem structure we are able to generate good quality upper bounds to the optimal value early in the algorithm, and the discrete do- main is used to guide us eciently in the search of … Read more

Risk aversion in multistage stochastic programming: a modeling and algorithmic perspective

We discuss the incorporation of risk measures into multistage stochastic programs. While much attention has been recently devoted in the literature to this type of model, it appears that there is no consensus on the best way to accomplish that goal. In this paper, we discuss pros and cons of some of the existing approaches. … Read more

Chance-constrained problems and rare events: an importance sampling approach

We study chance-constrained problems in which the constraints involve the probability of a rare event. We discuss the relevance of such problems and show that the existing sampling-based algorithms cannot be applied directly in this case, since they require an impractical number of samples to yield reasonable solutions. Using a Sample Average Approximation (SAA) approach … Read more