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I've just began attempting chapter 1 in a linear optimization book and it's fairly simple maximizing problems.

such as; maximize 3x + 2y such that 2x - y <= 6, 2x+ y <= 10 and x,y >= 0

Simple enough to find the feasible region, and where the maximum would occur algebraically.

In Complex Analysis we have the Maximum modulus principle, saying any bounded region achieves its maximum value on a boundary point... is it okay to say the same thing in these type of problems?

Also, is any mapping of the cost (maximizing 3x + 2y) going to be a 3-D graph or is there a way to make it 2-D?

The Pheromone Kid
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strugglingstudent
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  • You can't just "say" the same thing, but you can prove it. It's called the maximum principal and holds for interesting classes of functions, one of which is the harmonic functions. Any first order polynomial (in any number of variables) is harmonic (i.e. satisfies the Laplace equation), and therefore obtains its max and min at the boundary. – bjorne May 20 '14 at 21:16

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Regarding your first question, for linear programming, your statement is correct. Indeed, you will find the maxima of your optimization problem at the corners a convex polyhedron which is defined by the constraints of the problem, assuming that the polyhedron is bounded. However, this statement is not always true in optimization, nonlinear optimization problems can have their extreme points in the interior of the feasible region.

Regarding the second question, the mapping of the cost function will result in a 3D-graph.

The Pheromone Kid
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