Chapter One. Uncertain Linear Optimization Problems and their Robust Counterparts

In this chapter, we introduce the concept of the uncertain Linear Optimization problem and its Robust Counterpart, and study the computational issues associated with the emerging optimization problems. Recall that the Linear Optimization (LO) problem is of the form min x c T x + d : Ax ≤ b , (1.1.1) where x ∈ R n is the vector of decision variables, c ∈ R n and d ∈ R form the objective, A is an m × n constraint matrix, and b ∈ R m is the right hand side vector. Clearly, the constant term d in the objective, while affecting the optimal value, does not affect the optimal solution, this is why it is traditionally skipped. As we shall see, when treating the LO problems with uncertain data there are good reasons not to neglect this constant term. The structure of problem (1.1.1) is given by the number m of constraints and the number n of variables, while the data of the problem are the collection (c, d, A, b), which we will arrange into an (m + 1) × (n + 1) data matrix D = c T d A b. Usually not all constraints of an LO program, as it arises in applications, are of the form a T x ≤ const; there can be linear " ≥ " inequalities and linear equalities as well. Clearly, the constraints of the latter two types can be represented equivalently by linear " ≤ " inequalities, and we will assume henceforth that these are the only constraints in the problem. Typically, the data of real world LOs (Linear Optimization problems) is not known exactly. The most common reasons for data uncertainty are as follows: • Some of data entries (future demands, returns, etc.) do not exist when the problem is solved and hence are replaced with their forecasts. These data entries are thus subject to prediction errors; • Some of the data (parameters of technological devices/processes, contents associated with raw materials, etc.) cannot be measured exactly – in reality their values drift around the measured " nominal " values; these data are subject to measurement errors; • Some of the decision variables (intensities with which we intend to use various technological processes, parameters of physical devices we are designing, etc.) cannot be implemented exactly as computed. The resulting implementation errors are equivalent to appropriate artificial data uncertainties. Indeed, …