What do you understand by constrained optimisation? Let the utility function be U (X, Y) = X2 + Y. Price of commodity X is Px = 2 and price of commodity Y is Py = 1. The income is M = 200. Find out the equilibrium quantities of X and Y, given the constraint M = PxX+ PyY.

Constrained optimization refers to the optimization of a function subject to one or more constraints. In other words, it involves finding the maximum or minimum of a function, while ensuring that the solution satisfies certain constraints. Constrained optimization problems are commonly encountered in various fields, including economics, engineering, finance, and operations research.

The optimization problem can be formulated mathematically as follows: given a function f(x), the goal is to find the values of x that maximize or minimize f(x), subject to one or more constraints. The constraints can take different forms, such as equalities or inequalities, and may involve one or more variables.

For example, consider the following optimization problem:

minimize f(x) = x^2 + y^2

subject to the constraints: x + y = 1 x, y >= 0

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