In the popular computer game League of Legends, a player’s Effective Health, E, when defending against physical damage
Question A – Constrained Optimisation
In the popular computer game League of Legends, a player’s Effective Health, E, when defending against physical damage is given by E = H(1 + 0.01A) where H and A denote the player’s chosen units of health and armor, respectively. Furthermore, a unit of health costs 3 gold pieces and a unit of armor costs 18 gold pieces. If you have 3600 gold pieces to spend and you need to maximise the effectiveness E of your health and armor to survive as long as possible against enemy attacks, how much of each should you buy? A1. Write down the Lagrange function associated with this optimisation problem. A2. Solve the problem using the Lagrange Multiplier method (show your work). A3. Thirty minutes into the game, you have 2000 health and 50 armor. You have 1800 gold pieces to spend, and again health costs 3 gold per unit and armor costs 18 gold per unit. If the goal is still to maximise the effectiveness E of your resulting health and armor (after purchase), how much of each should you buy with your 1800 gold pieces (show your work).
Hint
Constrained optimization-a multivariable function can be maximize (or minimize) by Lagrangian expressionf(x,y,…)and g(x,y,…) = cthen follow these steps:Step 1: Initiate a new variable λ and new function L as follows: L(x,y,…,λ) = f(x,y,…) − λ(g(x,y,…) − c)L is called the "Lagrangian", and the new variable λ is termed&...
Constrained optimization- a multivariable function can be maximize (or minimize) by Lagrangian expression f(x,y,…) and g(x,y,…) = c then follow these steps: Step 1: Initiate a new variable λ and new function L as follows: L(x,y,…,λ) = f(x,y,…) − λ(g(x,y,…) − c) L is called the "Lagrangian", and the new variable λ is termed as a "Lagrange multiplier"
Step 2:critical points of L can be found by setting L equal to the zero vector. Step 3: it will look like (x0, y0,…,λ0). Plug each one into f. it will give the greatest (or smallest) value that will be maximum (or minimum) point you are seeking.
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