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<math>\min f(x, y(1), ..., y(N)) + wP(z(1), ..., z(N)): Ax=b, x \le 0,</math>
<math>\ B(s)x + C(s)y(s) + z(s) = e(s),</math> and <math>y(s) \ge 0,\, \forall s = 1,...,N,</math>
where f is a function that measures the cost of the policy, P is a penalty function, and w > 0 (a parameter to trade off the cost of infeasibility). One example of f is the expected value: f(x, y) = cx + Sum_s{d(s)y(s)p(s)}, where p(s) = probability of scenario s. In a worst-case model, f(x,y) = Max_s{d(s)y(s)}. The '''penalty function''' is defined to be zero if (x, y) is feasible (for all scenarios) -- i.e., P(0)=0. In addition, P satisfies a form of monotonicity: worse violations incur greater penalty. This often has the form P(z) = U(z^+) + V(-z^-) -- i.e., the "up" and "down" penalties, where U and V are strictly increasing functions.
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