Convex PolyHedron and Mathematical Optimization

Maximum or minimum value of a polynomial can be derived using calculus or linear algebra.
A convex polyhedron can be thought of as a n dimensional object in a bounded plane.
Algorithms such as the Simplex are based on traversing the polyhedron from vertex to vertex using certain methods in order to determine the extreme points.
There are other methods that are based on traversing the polyhedron by cutting through the polyhedron. These methods use the steepest gradient to determine incrementally the path to traverse to determine the extreme points and are called as interior point methods.
Convex polyhedron are geometric structures that have a definite maximum or minimum.

So, finding out an extreme point is very easy using a combination of calculus and linear algebra. When the polyhedron is characterized by straight lines the paradigm of convex optimization is termed as Linear Programming.

The polyhedron can also be quadratic that provides some simple minimums.
An example of quadratic polyhedron is a parabola which has a minimum point no matter what.
Random algorithms do not follow any rules to locate the extreme points of the polyhedron but use some random ascent and descent methods. So, how can business problems be formulated. Let us take the example of a factory that is producing two products. Product A yields P1 dollars of profit and Product B yields P2 dollars of profit.
Product P1 uses H1 hours of labor/unit produced and Product P2 uses H2 hours of labor/unit produced. Now we need to determine the optimum production mix that would maximize the profits given that only L hours of labor is available per week.
In addition, we are also given that we can produce only a maximum of M1 of product 1 and M2 units of Product 2 per week based on the demand.
Mathematically this reduces to some equations.
The objective function becomes Maximize: P1Q1 + P2Q2 ( 1) Subject To: Q1H1 + Q2H2 < L (2) Q1 < M1 (3) Q2 < M2 ( 4).

By plotting equations 2, 3 and 4 on a graph sheet one can see that it would form a two dimensional object. Since an extreme point cannot lie inside the two dimensional polyhedron the maximum value of the straight line (1) will be on the vertexes of the bounded two dimensional polyhedron. One can inspect all the edges using a brute force method and determine the maximum value of 1. On the contrary one can also use some special techniques to arrive at the solution by not traversing via all the edges.

This was essentially what we discussed in the first two paragraphs of this article. Analyzing properties of convex polyhedron such as volume, finding out the vertexes of the polyhedron can provide us with many interesting solutions in optimizing many business problems.
In real life business scenarios the number of edges can go to as many as 10000 and the number of dimensions can also be as large as 1000. There is a lot of research underway to determine the best possible method to compute the volume of a polyhedron or determine the fastest way to determine the maximum or minimum value of a function given that the solution lies on the vertexes of a convex polyhedron.