Scaling, Proximity, and Optimization of
Integrally Convex Functions111The extended abstract of this paper is included in the Proceedings
of the 27th International Symposium on Algorithms and Computation
(ISAAC), Sydney, December 12–14, 2016.
Leibniz International Proceedings in Informatics (LIPIcs),
64 (2016), 57:1–57:12,
In discrete convex analysis, the scaling and proximity properties for the class of L-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of variables, we show here that the scaling property only holds when , while a proximity theorem can be established for any , but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.
The proximity-scaling approach is a fundamental technique in designing efficient algorithms for discrete or combinatorial optimization. For a function in integer variables and a positive integer , called a scaling unit, the -scaling of means the function defined by . A proximity theorem is a result guaranteeing that a (local) minimum of the scaled function is close to a minimizer of the original function . The scaled function is simpler and hence easier to minimize, whereas the quality of the obtained minimizer of as an approximation to the minimizer of is guaranteed by a proximity theorem. The proximity-scaling approach consists in applying this idea for a decreasing sequence of , often by halving the scale unit . A generic form of a proximity-scaling algorithm may be described as follows, where denotes the -size of the effective domain and denotes the proximity bound in -distance for .
|S0:||Find an initial vector with , and set .|
|S1:||Find an integer vector with that is a (local) minimizer of|
|, and set .|
|S2:||If , then stop ( is a minimizer of ).|
|S3:||Set , and go to S1.|
The algorithm consists of scaling phases. This approach has been particularly successful for resource allocation problems [8, 9, 10, 16] and for convex network flow problems (under the name of “capacity scaling”) [1, 14, 15]. Different types of proximity theorems have also been investigated: proximity between integral and real optimal solutions [9, 31, 32], among others. For other types of algorithms of nonlinear integer optimization, see, e.g., .
In discrete convex analysis [22, 23, 24, 25], a variety of discrete convex functions are considered. A separable convex function is a function that can be represented as , where , with univariate discrete convex functions satisfying for all .
A function is called integrally convex if its local convex extension is (globally) convex in the ordinary sense, where is defined as the collection of convex extensions of in each unit hypercube with ; see Section 2 for precise statements.
A function is called L-convex if it satisfies one of the equivalent conditions in Theorem 1.1 below. For , and denote the vectors of componentwise maximum and minimum of and , respectively. Discrete midpoint convexity of for means
where and denote the integer vectors obtained by componentwise rounding-up and rounding-down to the nearest integers, respectively. We use the notation and for the -th unit vector , with the convention .
For the following conditions, (a) to (d), are equivalent:222-valued functions are treated in [4, Theorem 3], but the proof is valid for -valued functions.
(a) is integrally convex and submodular:
(b) satisfies discrete midpoint convexity (1.1) for all .
(c) satisfies discrete midpoint convexity (1.1) for all with , and the effective domain has the property: .
(d) satisfies translation-submodularity:
A simple example to illustrate the difference between integrally convex and L-convex functions can be provided in the case of quadratic functions. Indeed, for an symmetric matrix and a vector , the function is integrally convex whenever is diagonally dominant with nonnegative diagonal elements, i.e., for . On the other hand, is L-convex if and only if it is diagonally dominant with nonnegative diagonal elements and for all [23, Section 7.3].
A function is called M-convex if it satisfies an exchange property: For any and any , there exists such that
where, for , and . It is known (and easy to see) that a function is separable convex if and only if it is both -convex and -convex.
Integrally convex functions constitute a common framework for discrete convex functions, including separable convex, -convex and -convex functions as well as -convex and -convex functions , and BS-convex and UJ-convex functions . The concept of integral convexity is used in formulating discrete fixed point theorems [11, 12, 35], and designing solution algorithms for discrete systems of nonlinear equations [17, 34]. In game theory the integral concavity of payoff functions guarantees the existence of a pure strategy equilibrium in finite symmetric games .
The scaling operation preserves -convexity, that is, if is -convex, then is -convex. -convexity is subtle in this respect: for an -convex function , remains -convex if , while this is not always the case if .
Here is an example to show that -convexity is not preserved under scaling. Let be the indicator function of the set . Then is an -convex function, but (= with ), being the indicator function of , is not -convex. This example is a reformulation of [23, Note 6.18] for M-convex functions to -convex functions.
It is rather surprising that nothing is known about scaling for integrally convex functions. Example 1.1 does not demonstrate the lack of scaling property of integrally convex functions, since above is integrally convex, though not -convex.
As for proximity theorems, the following facts are known for separable convex, -convex and -convex functions. In the following three theorems we assume that , is a positive integer, and . It is noteworthy that the proximity bound is independent of for separable convex functions, and coincides with , which is linear in , for -convex and -convex functions.
Suppose that is a separable convex function. If for all , then there exists a minimizer of with .
The statement is obviously true if . Then the statement for general follows easily from the fact that is a minimizer of if and only if, for each , is a minimizer of . ∎
Suppose that is an -convex function. If for all , then there exists a minimizer of with .
Suppose that is an -convex function. If for all , then there exists a minimizer of with .
Based on the above results and their variants, efficient algorithms for minimizing -convex and -convex functions have been successfully designed with the proximity-scaling approach ([18, 20, 21, 23, 30, 33]). Proximity theorems are also available for -convex and -convex functions  and L-convex functions on graphs [6, 7]. However, no proximity theorem has yet been proved for integrally convex functions.
The new findings of this paper are
A “box-barrier property” (Theorem 2.6), which allows us to restrict the search for a global minimum of an integrally convex function;
As a consequence of our proximity and scaling results, we derive that:
When is fixed, an integrally convex function can be minimized in time by standard proximity-scaling algorithms, where denotes the -size of .
This paper is organized as follows. In Section 2 the concept of integrally convex functions is reviewed with some new observations and, in Section 3, their scaling property is clarified. After a preliminary discussion in Section 4, a proximity theorem for integrally convex functions is established in Section 5. Algorithmic implications of the proximity-scaling results are discussed in Section 6 and concluding remarks are made in Section 7.
2 Integrally Convex Sets and Functions
For the integer neighborhood of is defined as
For a function the local convex extension of is defined as the union of all convex envelopes of on as follows:
where denotes the set of coefficients for convex combinations indexed by :
If is convex on , then is said to be integrally convex . A set is said to be integrally convex if the convex hull of coincides with the union of the convex hulls of over , i.e., if, for any , implies . A set is integrally convex if and only if its indicator function is an integrally convex function. The effective domain and the set of minimizers of an integrally convex function are both integrally convex [23, Proposition 3.28]; in particular, the effective domain and the set of minimizers of an L- or M-convex function are integrally convex.
For , integrally convex sets are illustrated in Fig. 1 and their structure is described in the next proposition.
A set is an integrally convex set if and only if it can be represented as for some and .
Consider the convex hull of , and denote the (shifted) unit square by , where . Let be an integrally convex set. It follows from the definition that for each . Obviously, can be described by (at most four) inequalities with and , where . Since is the union of sets , can be represented as by a subfamily of the inequalities used for all . Then we have . Converesly, integral convexity of set represented in this form for any and is an easy consequence of the simple shape of the (possibly unbounded) polygon , which has at most eight edges having directions parallel to one of the vectors , , , . ∎
We note that in the special case where all inequalities defining in Proposition 2.1 satisfy the additional property , the set is actually an -convex set [23, Section 5.5], which is a special type of sublattice .
A subtle point in Proposition 2.1 is explained here. In Proposition 2.1 we do not mean that the system of inequalities for describes the convex hull of . That is, it is not claimed that holds. For instance, is an integrally convex set, which can be represented as the set of integer points satisfying the four inequalities: , , , and . These inequalities, however, do not describe the convex hull , which is the line segment connecting and . Nevertheless, it is true in general (cf. the proof of Proposition 2.1) that the convex hull of an integrally convex set can be described by inequalities of the form of with and . For we can describe by adding two inequalities and to the original system of four inequalities. The present form of Proposition 2.1, avoiding the convex hull, is convenient in the proof of Proposition 3.1.
If a set is integrally convex, then for all points , the set
is contained in S.
Let be represented as in Proposition 2.1 and let . Then we clearly have . The claim follows by observing that coincides with one of , , , , , , according to the values of . ∎
Note that is integrally convex by Proposition 2.1, and that, by the above corollary, any integrally convex set containing must contain . Thus is the smallest integrally convex set containing .
Integral convexity is preserved under the operations of origin shift, permutation of components, and componentwise (individual) sign inversion. For later reference we state these facts as a proposition.
Let be an integrally convex function.
(1) For any , is integrally convex in .
For any permutation of ,
is integrally convex in .
(3) For any , is integrally convex in .
The claims (1) to (3) follow easily from the definition of integrally convex functions and the obvious relations: , , and . ∎
Integral convexity of a function can be characterized by a local condition under the assumption that the effective domain is an integrally convex set. The following theorem is proved in  when the effective domain is an integer interval (discrete rectangle). An alternative proof, which is also valid for the general case, is given in Appendix A.
Theorem 2.4 ([2, Proposition 3.3]).
Let be a function with an integrally convex effective domain. Then the following properties are equivalent:
(a) is integrally convex.
(b) For every with we have
Let be an integrally convex function and . Then is a minimizer of if and only if for all .
Theorem 2.6 (Box-barrier property).
Let be an integrally convex function, and let and , where . Define
and . Let . If for all , then for all .
Let , for which we have . For a point , the line segment connecting and intersects at a point, say, . Then its integral neighborhood is contained in . Since the local convex extension is a convex combination of the ’s with , and for every , we have . On the other hand, it follows from integral convexity that for some with . Hence , and therefore, . ∎
Corollary 2.7 (Hyperplane-barrier property).
Let be an integrally convex function. Let , , and let be an integer with . If and for all with , then for all with .
We denote the sets of nonnegative integers and positive integers by and , respectively. For we write for . For vectors with , denotes the interval between and , i.e., , and the integer interval between and , i.e., .
3 The Scaling Operation for Integrally Convex Functions
In this section we consider the scaling operation for integrally convex functions. Recall that, for and , the -scaling of is defined to be the function given by .
When , integral convexity is preserved under scaling. We first deal with integrally convex sets.
Let be an integrally convex set and . Then is an integrally convex set.
Next we turn to integrally convex functions.
Let be an integrally convex function and . Then the scaled function is integrally convex.
Proposition 3.3 (Basic parallelogram inequality).
For an integrally convex function we have
by integral convexity of . By adding these inequalities for with and , we obtain (3.4). Note that all the terms involved in these inequalities are finite, since for all and . ∎
If , is not always integrally convex. This is demonstrated by the following example.
In view of the fact that the class of -convex functions is stable under scaling, while this is not true for the superclass of integrally convex functions, we are naturally led to the question of finding an intermediate class of functions that is stable under scaling. See Section 7 for this issue.
4 Preliminary Discussion on Proximity Theorems
Let and . We say that is an -local minimizer of (or -local minimal for ) if for all . In general terms a proximity theorem states that for there exists an integer such that if is an -local minimizer of , then there exists a minimizer of satisfying , where is called the proximity distance.
Before presenting a proximity theorem for integrally convex functions in Section 5, we establish in this section lower bounds for the proximity distance. We also present a proximity theorem for , as the proof is fairly simple in this particular case, though the proof method does not extend to general .
4.1 Lower bounds for the proximity distance
The following examples provide us with lower bounds for the proximity distance. The first three demonstrate the tightness of the bounds for separable convex functions, -convex and -convex functions given in Theorems 1.2, 1.3 and 1.4, respectively.
Example 4.1 (Separable convex function).
Let for and define , which is separable convex. This function has a unique minimizer at , whereas is -local minimal and . This shows the tightness of the bound given in Theorem 1.2.