Convex optimization and the smallest ball problem

The Smallest Ball Problem is a famous problem in mathematics that was proposed by James Joseph Sylvester. In the past, many algorithms to solve this problem were founded but there was only a limited number of researches that focused on the rigorous mathematical proof of this problem. Thus, the goal of this essay is to provide a rigorous proof of the claim that the smallest enclosing ball must exist and it is unique. In this essay, the Smallest Ball Problem will be converted into a convex optimization problem and the result that the smallest enclosing ball exists and is unique can be proved by proving the optimal solution of this programming problem exists. The meaning of this research is to give a theoretically mathematical proof of the Smallest Ball Problem so that it can tell the algorithms to solve this problem can always work. Thus, it can also ensure the effectiveness of all the related algorithms.


Background
The Smallest Ball Problem is also called the Smallest Enclosing Ball Problem.It is a geometry problem of finding the ball that can cover all the given points in the real vector space with its radius reaches the minimum value [1].In 1857, this problem was proposed by James Joseph Sylvester, who was an mathematician from England.In particular, when considering the Euclidean plane, this problem becomes the smallest circle problem, and it is an instance of the 1-center problem.It is a problem targeted at finding the minimum of the longest distance that a customer should travel to reach the new facility [2].With the development of computer science, researchers found that the worst runtime of solving this problem is O(n).

Current Research
Now, there are many methods to find the answer to the Smallest Ball Problem.In fact, mathematicians found that there exists algorithms to solve such problem in linear time and there are two main algorithms that can help to find the solution, called Megiddo's algorithm and Welzl's algorithm.Megiddo ) + cn where t(n) = 16cn [3].Indeed, this algorithm is quite complicated and it cannot avoid solving a similar problem that the enclosing ball's center must be located on a given line twice to finish the reduction.However, this algorithm can be optimized and indeed, it can reduce the unnecessary points.Another important algorithm is Welzl's algorithm whose main idea is recursion.First of all, it randomly and uniformly picks a point p from P, which consists all the points, and finds the smallest ball containing the rest of the points in point set P. If the result also encloses p, then it is actually the smallest ball that is targeted.Otherwise, the point p is on the boundary of the resulting ball and continue to do the recursion with the additive parameter of the point set Q that consists of all the points on the boundary.The recursive steps are stopped until P becomes empty and the smallest ball can be found by Q, which is the set of all points on the boundary [4].

Relating problem
In fact, there are many geometry problems that can be solved by optimization, one is called the Chebyshev Center Problem.This problem shares many similarities with the Smallest Ball Problem since it also focuses on finding the smallest ball.The difference is that the Chebyshev Center Problem is finding the inscribed ball.The Chebyshev center of a bounded set P with non-empty interior is the center of the ball with the minimum radius that can enclose P, alternatively, it also has an equivalent definition that is the center of the inscribed ball of P with the maximum value of its radius [5].In fact, this geometry problem has an equivalent form to a linear programming problem as follows, here, P is the intersection of finitely many hyperplanes [6]:

Motivation for the research
Although researchers in the past have created many algorithms to compute the ball satisfying the condition of the Smallest Ball Problem, most of the research in this field does not focus on the rigorous mathematical proof of the statement that such ball exists and it is unique.Thus, this essay targets at providing a rigorous proof of this problem to state that this conclusion is true.To achieve this goal, firstly, it is important to know what is convex programming problem since the Smallest Ball Problem can actually be converted to such a problem and then, several important conclusions will be discussed and used to solve the targeted problem.

The Smallest Ball Problem and its mathematical modelling
Here is the description of the Smallest Ball Problem: Given a point set of size n  = { 1 ,  2 , … ,   }, where   ∈   , 1 ≤  ≤ ,      −     .This problem asks to find a ball with the smallest radius that can cover all the points in .To solve this problem, the first step is converting it into a convex programming problem.The following convex programming problem is corresponding to the Smallest Ball Problem: Let  ∈  × (), ℎ ℎ ℎ  ℎ    is all the components of the vector represented by the point   .
Considering such optimization problem with the objective function f ,  ≥ 0 [7].Obviously, the objective function f(x) =

Method of solving the Smallest Ball Problem
In fact, the following two claims can be made and actually it is the result of this optimization problem.
Claim1: This optimization problem has an optimal solution  * ; Claim 2: There exists  * , such that  * =  * ℎ      * .Moreover, the ball with center  * ,  ( * ) is the unique ball that can satisfy all the conditions of the Smallest Ball Problem.
By the above two claims, this problem can be solved since claim2 can determine this ball.For the purpose of proving these two claims, several conclusions will be applied including an amazing proposition called the Karush-Kuhn-Tucker multipliers.This essay will give a rigorous proof of the targeted problem including all the theorems and lemmas needed to solve this problem and these, including the two claims, will be proved in the third part of this essay.

Solving the Smallest Ball Problem
In this part, first of all, the concept of convex optimization will be given and two important theorems including the famous Karush-Kuhn-Tucker conditions will be introduced and proven.With the method of convex optimization and the help of the two theorems, it is sufficient to give a rigorous mathematical proof of this problem.In the process of solving the targeted problem, another lemma relating to the Smallest Ball Problem will also be presented and proved.

Introduction to convex programming problem
Firstly, it is important to know what is convex optimization.
A convex program is a problem whose goal is to minimize a convex function.It has the equational form as follows: Minimize f(x) Subject to Ax=b, x≥ 0 where A∈  × (),  ∈   ,  :   →      Actually, since calculus should be used so the assumption that f is differentiable on   can be made.By linear approximation, such inequalities can be got: Then an important conclusion, which is the Theorem1 in Chapter 3.2, can be found in this inequality.

Solving the Smallest Ball Problem
Theorem 1 and Karush-Kuhn-Tucker conditions are proved, and by now it is able to solve the Smallest Ball Problem using these two amazing facts.However, before coming to the problem, there is a lemma related to the problem that should be proved.Lemma: Let S={ ball and the related problem called Chebyshev Center.Then, in Chapter 2 converting such a geometry problem into the form of a programming problem and introducing the general idea of the research topic.Chapter3, first, introduces the definition of a convex programming problem and then gives proof to several important conclusions (Theorem 1 and Theorem 2) and a lemma that is related to the Smallest Ball Problem.In the last part of the Chapter3, the statement that the ball can cover all the points with the smallest radius is proved rigorously.Moreover, at the end of this essay, several ideas to do deeper research in this field are discussed.In general, this essay focuses on the proof of the Smallest Ball Problem, instead of its algorithm, which has been done by many researchers.So this essay can solve this famous problem mathematically which is the aspect that gained less attention from researchers.
− ∑       =1  is a convex function since it is in quadratic form.Here, our objective function is the characterization of the ball.So the Smallest Ball Problem can actually be converted into an optimization problem.