Linear programming (simplex/...) is a special kind of optimization in the sense that the problem can be simplified to a few vertices due to the convexity/regularity/predictability of linearity.
Separation theory translates the spacial relationship of keeping a certain distance from something and only to one side, to the algebraic expression using a notion of the linear algebraic vectors (depicting the side) and a notion of order (depicting the distance). In other words, optimization can be directly translated to spacial limit searching if ever optimization is possible (somewhat convex).
Nonlinear programming (KT/...) comes from the the concept of simplification of optimization problems in linear programming according to geometrically intuitive properties. It is in contrast to the second-order calculus approach which also come from geometrical concepts but used a more complicated representation of convexity thus making the solution process less preferred. Such complication also imposed unnecessary restrictions on the types of solvable problems, especially in terms of the effectiveness of constraints, where in settings such as the Lagrangian problem constraints have to be in equality in order for the first order conditions to be 'conveniently' fulfilled.
The envelope theorem states that the total effect on equilibrium state of the change of a parameter in the optimization process is the same as if no optimization process is gone through. Varian explained it more intuitively in his Microeconomic Analysis, page 45 (profit function).
Separation theory translates the spacial relationship of keeping a certain distance from something and only to one side, to the algebraic expression using a notion of the linear algebraic vectors (depicting the side) and a notion of order (depicting the distance). In other words, optimization can be directly translated to spacial limit searching if ever optimization is possible (somewhat convex).
Nonlinear programming (KT/...) comes from the the concept of simplification of optimization problems in linear programming according to geometrically intuitive properties. It is in contrast to the second-order calculus approach which also come from geometrical concepts but used a more complicated representation of convexity thus making the solution process less preferred. Such complication also imposed unnecessary restrictions on the types of solvable problems, especially in terms of the effectiveness of constraints, where in settings such as the Lagrangian problem constraints have to be in equality in order for the first order conditions to be 'conveniently' fulfilled.
The envelope theorem states that the total effect on equilibrium state of the change of a parameter in the optimization process is the same as if no optimization process is gone through. Varian explained it more intuitively in his Microeconomic Analysis, page 45 (profit function).
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