How to minimize results using convex optimization

Convex Optimization is a procedure to minimize the convex function over sets. The procedure increases efficiency and reduces time.

Convex Functions

Let's say that there is a function that maps values to a graph as follows (we've made the graph below):

The $Chord$ and $Curve$ is further simplified below for convexity:

In counter to this, we have the concave functions. Now let's discuss convex optimization in detail.

Convex optimization

Convex optimization tends to minimize the objective functionthe function to be minimized and the main plotting function.

Convex sets

Because convex functions are minimized over convex sets, hence they're discussed here. Look at the image below:

A convex set will contain all the points between Point 1 and Point 2, meaning that the chord will lie in the pinkish area. That area contains all the values and points of that graphical plot.

How to find optimal solutions

There can be many feasible solutions to a problem where we need to find a convex set. This means that the condition to be satisfied is just that the points should belong to the domain of the total points (within it). These feasible solutions can be written in a set as follows:

Now that we have limited the set, we just need to find the optimal solution from this set. What we now need to do is just to find the local minimum from the sets and obtain the optimum solution.

The optimal variable or point that is minimum should not just fulfill the main convex function but also the constraint functions, and that is exactly what is shown by the formula below:

How to solve the convex optimization problem

There are some ways we can use to optimize such problems: