PminPmax

PminPmax is a term used in optimization to describe a min–max decision principle in which a quantity is minimized with respect to a parameter and the worst‑case outcome is taken with respect to an adversarial or uncertain variable. The notation PminPmax often denotes the outer minimization over a parameter space P and the inner maximization over a secondary space S: min_{p ∈ P} max_{s ∈ S} f(p, s). The resulting value represents the best guaranteed performance against the most adverse scenario within S for the chosen parameter p.

In this framing, p is selected to minimize the maximum value of f over all s in S. This contrasts with other orderings of quantifiers, such as max_{s} min_{p} f(p,s), and is commonly used in robust optimization, control, and decision making under uncertainty.

Under mild regularity conditions (e.g., P and S are compact, f is continuous), the optimal p exists and the inside maximum is well defined. If f is convex in p for every s and concave in s for every p, a saddle point may exist and duality results can facilitate computation. In general, however, PminPmax problems are hard, often NP-hard, and require specialized algorithms.

Applications include designing robust controllers, risk-averse investment strategies, and machine learning models that resist adversarial inputs. Variants include PminPmax with different inner/outer orders or with additional constraints, and connections to minimax and robust optimization literature.

Example: consider f(p,s) = (p − s)^2 with p in [0,1] and s in [0,1]. The PminPmax value is min_{p in [0,1]} max_{s in [0,1]} (p − s)^2, which occurs at p = 0.5 with value 0.25.