Calculus Volume 3 · 免费预览
拉格朗日乘数
约束优化在约束条件 g 等于 c 下求 f 的极值，其驻点为两梯度平行之处，即梯度 f 等于 lambda 乘以梯度 g；这一乘数方程组把约束问题转化为方程求解问题。
1. The Lagrange condition
The Lagrange condition
2. The multiplier's meaning
The multiplier's meaning
3. The general case and the boundary
The general case and the boundary
4. When the method applies
When the method applies
5. Lagrange condition
At a constrained extremum of f subject to g = c, the gradients satisfy which equation?
6. Tangency geometry
Geometrically, the Lagrange condition says the level curve of f is which relation to the constraint curve g = c at the extremum?
7. Multiplier meaning
The multiplier lambda in the Lagrange method equals which derivative?
8. Constraint gradient condition
The Lagrange method requires the constraint's gradient to be nonzero, called which condition?
9. Utility maximization setup
Maximizing utility U(x, y) subject to a budget p_x x + p_y y = B is which kind of problem?
10. Multiplier symbol
The Lagrange multiplier is conventionally which Greek letter? Answer with the letter name spelled out.
11. System equation count
For f(x, y) subject to g(x, y) = c, the Lagrange system has two gradient equations plus the constraint. As a number, how many equations? Answer with a number.
12. Shadow price field
The shadow price interpretation dominates which discipline's constrained optimization? Answer with one word.
13. Failure of regularity
Where the constraint's gradient vanishes, the Lagrange method fails, and the candidates there must be found how?
14. Several constraints factor
With two constraints, gradient f is a combination of the two gradients with how many multipliers? Answer with a number.