Optimization and Mathematical Programming

This algorithm of artificial intelligence keeps only a single state in memory but can get stuck on local maxima.

Linear Programming is indeed a mathematical optimization technique used to determine the best allocation of limited resources among competing activities. It maximizes or minimizes an objective function subject to constraints.

Linear programming is deterministic, not probabilistic. All parameters in an LPP (objective coefficients, constraint coefficients, RHS values) are assumed to be known with certainty. Stochastic programming handles probabilistic scenarios.

In an LPP, the number of decision variables and constraints are independent. You can have m constraints and n decision variables with no requirement that m = n. Simplex method works with m ≠ n.

The graphical method is specifically designed for LPPs with exactly two decision variables. It plots the feasible region and objective function on a 2D coordinate plane. For three+ variables, use simplex or other methods.

A redundant constraint does not bind at the optimal solution point. Since it lies outside the active feasible region boundary, removing it doesn't change the feasible region or affect the optimal solution value.

Artificial variables are necessary in both maximization and minimization LPP when constraints are of the ≥ or = type. They help find an initial basic feasible solution when one doesn't exist naturally.

The simplex method requires all decision variables to be non-negative. Unrestricted variables (which can be positive or negative) must be converted by expressing each as the difference of two non-negative variables: x = x' - x''.

The minimum ratio test (least non-negative ratio) determines which variable reaches zero first as the entering variable increases. This selection ensures that feasibility is maintained - the chosen departing variable becomes non-basic while all other basic variables remain non-negative.

In optimization and search, the heuristic function is often referred to as the objective function because it defines the goal or the 'cost' that the algorithm aims to minimize or maximize.

Steepest-Ascent Hill-Climbing evaluates all possible successors of the current state and chooses the one that provides the greatest improvement, rather than just taking the first better move found.

Constraint satisfaction problems involve finding values for variables that satisfy a set of constraints, operating within a space defined by those constraints.