- Open Access
Midconvexity for finite sets
© Tabor et al.; licensee Springer 2013
- Received: 19 June 2012
- Accepted: 10 December 2012
- Published: 11 February 2013
Let X be a finite subset of a real vector space. We study Jensen-type convexity on subsets of X. In particular for subsets of X, we introduce the definition of X-midconvex sets. We show that such a notion corresponds well to the classical notion of a convex set. Moreover, we prove that a function X-midconvex set is a midconvex hull of all its extremal points. Other analogues of some classical results are also given. At the end we present an algorithmic approach to finding the midconvex hull of a given set.
- midconvex set
- midconvex function
- midconvex hull
Thanks to the growing role of computers, one can recently notice an increasing interest in discrete (finite) sets in many parts of mathematics. In particular, one has to mention the recent development of convexity on subsets of .
Therefore, there appears a natural need to introduce and develop (mid-)convexity on finite sets. Since we are going to study finite subsets of a real vector space, we may restrict ourselves to finite subsets of . Since the middle point of an interval is one of the most important geometrical notions, we start our investigations with the simplest and earliest version of convexity-midconvexity.
The main difference between our approach and that of [1–3] is that we restrict our considerations to X, that is, we treat X as a space. Hence, our notions have relative character since we study convexity on the subsets of X. One can say that we investigate midconvexity on a restricted domain.
Now, we briefly describe the contents of the paper. First of all, we propose a proper definition of a midconvex set for finite sets. By ‘proper’ we mean here that we are looking for the notion which has possibly many properties analogous to those of convex sets in . We consider two different definitions of midconvexity. The first one, which is intuitively very similar to the classical definition, occurs to have weak properties. The other one, which we call function midconvexity, at a first glance seems a little bit artificial, but it corresponds well to the classical definition. It is based on the observation that in a closed set V is convex if and only if it is the preimage of the set for some convex function .
Next, we study the properties of function midconvexity. In particular in Section 3, we prove an analogue of the theorem stating that a compact convex set in is the convex hull of its extremal points. We introduce also the notion of a convex function on a finite domain. We prove that the maximum of such a convex function is achieved at an extreme point of its domain.
In the whole paper, we assume that X is a finite subset of . By we denote the set of nonnegative integers and by the set of nonnegative real numbers. We adapt the classical definitions of a midconvex set and a midconvex function to our setting .
Evidently, X and a singleton are X-midconvex sets. It is also obvious that the intersection of X-midconvex sets is X-midconvex.
We will need some of their properties with respect to the following definition of a midconvex set.
Proposition 2.3 Letbe an X-midconvex set, letand let, be X-midconvex functions. Thenand cf are X-midconvex functions.
One could expect that if W is X-midconvex and are such that , then . It occurs that this is not the case.
Example 2.4 Let , and . Then W is X-midconvex and , but .
Therefore, we need another definition of a midconvex set.
Let us present the idea which, in our opinion, leads to the ‘right’ definition. The observation that a closed set is convex if and only if there exists a convex function such that (the part ‘if’ of the statement is obvious; for the converse, take as f the distance from A) leads us to the following definition.
Clearly, , is X-midconvex. Hence, X is X-fmidconvex.
Observation 2.6 Every X-fmidconvex set is X-midconvex.
Hence . □
As one could expect, the implication converse to Observation 2.6 does not hold. To notice this (see Example 2.8), we will need the following result, which shows that fmidconvexity resembles classical convexity.
Proof The proof follows a similar approach from .
- 1.. Then , and we obtain that
. Then , and by reasoning analogous to case 1, we obtain a contradiction.
Now, we show that the implication converse to that from Observation 2.6 does not hold.
Example 2.8 We continue considerations from Example 2.4. Let , , , . Then , and hence . W is an X-midconvex set which is not X-fmidconvex. It shows also that an analogue of Theorem 2.7 for X-midconvex set does not hold.
Proposition 2.9 Letand letbe X-fmidconvex andbe-fmidconvex. Thenis X-fmidconvex.
Evidently, . We have to prove yet that f is X-midconvex. Consider arbitrary such that . We distinguish five cases.
We have proved that f is X-midconvex. □
Now, we define an X-fmidconvex hull of a given set . We begin with important remarks.
Proposition 2.10 Let, be X-fmidconvex subsets of X. Thenis X-fmidconvex.
By Proposition 2.3 f is X-midconvex. Furthermore, . This finishes the proof. □
Proposition 2.11 Let, be finite sets and letbe-fmidconvex andbe-fmidconvex. Thenis-fmidconvex.
We are going to define now an X-midconvex hull and an X-fmidconvex hull of a given set . We follow the classical definition of a midconvex hull .
Definition 2.12 Let .
The intersection of all X-midconvex sets containing A is called an X-midconvex hull of A and is denoted by .
The intersection of all X-fmidconvex sets containing A is called an X-fmidconvex hull of A and is denoted by .
One can directly verify that is X-midconvex. By Proposition 2.10, we obtain that the intersection of a finite family of X-fmidconvex sets is X-fmidconvex, and consequently, by the finiteness of X, we obtain that is X-fmidconvex. Consequently, the X-fmidconvex hull of A is the smallest X-fmidconvex set containing A.
The example below illustrates Definition 2.12 and shows that an X-midconvex hull and an X-fmidconvex hull of a given set are, in general, different.
We begin with the definition of an extremal point.
We denote the set of all extremal points of W by extW. As we know , by the Krein-Milman theorem, a compact convex set in is the convex hull of its extremal points. However, its analogue does not hold for X-midconvexity-just consider which is not the X-midconvex hull of the set of its extremal points.
In this section, we prove a version of the classical Krein-Milman theorem for function midconvexity. We begin with the following observation.
Observation 3.2 Ifis nonempty and finite, then the.
Proof Take a convex hull of W in . It is nonempty, convex and compact (because W is finite). Therefore,  it has an extremal point. □
We will need the notion of index.
We call the beginning and the end of the J-chain.
Condition (2) means that is the middle of and a certain point in W (or, in another words, that the symmetric point to with respect to belongs to W).
Proposition 3.4 Let. Then for each, there exists a J-chain in W beginning in extW and ending at x.
Proof Fix arbitrarily and denote by Z the set of points z of W such that there exists a J-chain in W beginning at z and ending at x. Obviously , and hence . By Observation 3.2, there exists . Let , , , be the J-chain with beginning at z and end at x.
We show that . Suppose, for an indirect proof, that . Since , there exist , such that . Clearly, and are J-chains starting at v and w, respectively, and with end at x. This implies that , which contradicts the assumption that . □
Let be nonempty. By Observation 3.2, . For by , we denote the length of the shortest J-chain in W beginning in extW and ending at x. It follows from Proposition 3.4 that such a J-chain exists.
is called the index of x and -the index of W.
Remark 3.5 It follows from the proof of Proposition 3.4 that if , where , then there exists an element such that . Therefore, if , , then for each , we can find such that . Hence, the set of indices of elements of W coincides with the set .
One can easily notice that if and only if .
We illustrate the above notations by a simple example.
Since p is an extremal point of P either or . Assume that . Then and since , we obtain that . Hence, , a contradiction. Thus p is an extremal point of W and hence , a contradiction. We have proved that . □
Now, we prove the main result of this section.
a contradiction. □
Theorem 3.9 Let. Ifis X-midconvex, then the maximum of f is achieved at an extreme point of W.
a contradiction. □
There is a problem to find a convenient procedure to determinate an X-fmidconvex hull for a given set . In this section, we present an algorithm for finding , which can be easily implemented in high level programming languages.
We start with the following auxiliary result.
there exists an X-midconvex functionsuch thatand.
Proof Observe first that multiplying a midconvex function by a positive real number, we do not destroy midconvexity.
Then , and is X-fmidconvex. By Proposition 2.10 we obtain that is X-fmidconvex. It follows from Definition 2.12 that . Thus .
and . □
Proposition 4.1 gives the way to determine . Namely, by condition (2) we can eliminate from X, step by step, elements of the set .
Observe that is finite as .
Let us now fix . We want to check if there exists an X-midconvex function such that and .
wheredenotes zero in (we will often omit the subscript m and write 0).
for all , .
where is the i th row of the matrix . This finishes the proof. □
Remark 4.3 To save time and memory in practical implementation, in Proposition 4.2 we take into account only one of the pairs and , since the inequalities and are equivalent.
f is midconvex,
This completes the proof since . □
We illustrate the above considerations by a simple example.
Further discussion of this example can be found in Example 4.6.
where y represents the vector of variables (to be determined). In our case . The system of inequalities denotes the constraints which specify a convex polygon, over which the objective function is to be optimized.
We still need to ‘insert’ additional information that (which is equivalent to , as we can divide f by ). In other words, we ask if we can find a function satisfying the previously defined systems of inequalities and such that . To do this, we put , where is such that , (according to (4)), and find a solution to the system of inequalities defined above.
Firstly, we check if 1 is in the X-fmidconvex hull of the set W. We put . It is easy to get the following solution: , and , , (this means that can be arbitrary, e.g., ) this solution is bounded, so . The same consideration shows that also . On the contrary, for 4 the value is unbounded so . Finally, we obtain that .
There are many methods for solving LP (checking whether a solution exists is just as difficult as finding a solution): simplex algorithm of Dantzig , criss-cross algorithm , projective algorithm of Karmarkar , etc.
However, the method for finding an X-fmidconvex hull presented in this section is not sufficiently efficient in practice because for each point of investigated space X, we solve LP. Moreover, if we use a classical simplex method, we can perform really badly since the worst-case complexity of the simplex method has exponential time.
Sample implementation of the algorithm described in this section with nice graphical interface prepared in Java programming language is available at http://www.ii.uj.edu.pl/~misztalk.
The first author was supported by National Centre of Science (Poland) Grant No. 2011/01/B/ST6/01887.
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