Conceptual question on optimization 
*

*What is the intuition and the physical meaning of the mathematical expression in convex optimization?

*When using optimization algorithms like particle swarm or genetic algorithm, do they have anything do to with convex optimization? Are the algorithms for convex optimization?
 A: Convex optimization problems form a subclass of optimization problems that we can currently solve quite efficiently, namely convex functions over convex sets. Historically, it was assumed that linearity or lack thereof was the deciding factor in the difficulty of an optimization problem but it turns out that convexity is a much better indicator of difficulty.


*

*Not entirely sure what you mean by the mathematical expression. I assume you are asking about the definition of a convex set. Intuitively, you can think of it like this: a set is convex if we can construct a straight line segment between any two points in the set and the whole line will also be in the set. Examples are circles, squares, triangles, $\ldots$. A horseshoe or banana shape does not meet this requirement.

*Evolutionary algorithms like PSO and GA are general purpose methods that can solve optimization problems with very few assumptions (you can throw almost anything at them and they will usually do a pretty good job). They do not require convexity (nor smoothness, nor continuity, $\ldots$). General purpose algorithms can be applied more broadly, but are also less efficient than convex optimization methods when dealing with a convex problem (slower convergence, little or no guarantees). Examples of convex methods include (sequential) quadratic programming and semidefinite programming.
