What is convex optimization

Optimization book

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1. If the optimization problem is convex, then any local minimizer is also a global minimizer.
Thus, if you can use gradient descent techniques, to solve any convex problem optimally. .
2. If the optimization problem is strongly convex, then if you can find a local minima, then that is also a unique global minima

Optimization book

1 Answer.
No, not all convex programs are easy to solve.
There are intractable convex programs.
Roughly speaking, for an optimization problem over a convex set X to be easy, you have to have some kind of machinery available (an oracle) which efficiently can decide if a given solution x is in X..

Why do we care about convex in optimization?

Because the optimization process / finding the better solution over time, is the learning process for a computer.
I want to talk more about why we are interested in convex functions.
The reason is simple: convex optimizations are "easier to solve", and we have a lot of reliably algorithm to solve._{Jan 25, 2018}.