Linear programming under vacuous and possibilistic uncertainty

Consider the following (standard) linear programming problem: maximise a real-valued linear function CT x defined for optimisation variables x in Rn that have to satisfy the constraints Ax ≤ B, x ≥ 0, where the matrices A, B, and C are independent random variables that take values a, b, and c in Rm×n, Rm and Rn, respectively. Using an approach we developed in previous work [3], the problem is first reduced to a constrained optimisation problem (co-problem) from which the uncertainties present in the description of the constraint are eliminated. The goal is to derive efficient solution techniques for this resulting co-problem. We investigate what results can be obtained for two types of uncertainty models for the random variables A, B, and C – vacuous previsions and possibility distributions [see, e.g., 1, 5] – and for two different optimality criteria – maximinity and maximality [see, e.g., 4]. In our poster, we will present the problem description and show illustrated solutions for the most interesting cases we have investigated. We consider three variants of our problem: (i) when there is no uncertainty about C (this exactly fits the approach in [3]), (ii) when there is no uncertainty about B, which reduces to variant (i) when considering the dual, and (iii) the general case, which we can convert to the following problem: maximise the real value λ such that Ax≤ B, CT x≥ λ and x≥ 0, and which is the subject of current research. We here focus on variant (i). For the different cases we studied, the co-problem and solution techniques derived are: • Vacuous model relative to a set A ⊆ Rm×n×Rm: – The maximin solution xm can be found by solving the linear programming problem argmaxx∈I cT x, where I := ⋂ (a,b)∈A {x ∈ Rn : ax≤ b} is the inner feasibility space. – The maximal solutions can be found by vertex enumeration [see, e.g., 2] of { x ∈ Rn : x ∈O and cT x≥ cT xm } , where O := ⋃ (a,b)∈A {x ∈ Rn : ax≤ b} is the outer feasibility space, which turns out to be convex. • Possibility distribution π on Rm×n×Rm with corresponding lower probability Pπ : – The maximin solution is given by argmaxx∈Rn ( cT x−L ) Pπ(Ax ≤ B) where L is a penalty for violating the constraints. When the possibility distribution π is unimodal then ( cT x−L ) Pπ(Ax≤ B) is unimodal too because of the linearity of the objective function, which allows us to find the maximin solution using a bisection method in which each step a linear programming problem must be solved. – We have not yet found an efficient way to calculate the maximal solutions. We can approximate the solutions when m and n are small enough.