Abstract
First, we introduce two new reformulation convexification based hierarchies called RTC and RSC for which the rank continuous relaxations are denoted by and , respectively. These two hierarchies are obtained using two different convexification schemes: term convexification in the case of the RTC hierarchy and standard convexification in the case of the RSC hierarchy. Secondly, we compare the strength of these two hierarchies. We will prove that (i) the hierarchy RTC is equivalent to the RLT hierarchy of Sherali-Adams, (ii) the hierarchy RTC dominates the hierarchy RSC, and (iii) the hierarchy RSC is dominated by the Lift-and-Project hierarchy. Thirdly, for every rank , we will prove that and where the sets and are convex, while and are two nonconvex sets with empty interior (all these sets depend on the convexification step). The first inclusions allow, in some cases, an explicit characterization (in the space of the original variables) of the RLT relaxations. Finally, we will discuss weak version of both RTC and RSC hierarchies and we will emphasize some connections between them.
1. Introduction
Let and be two integers. Let and be the two sets and , respectively. Let be a subset of representing the set of feasible solutions of a mixed integer linear program. The integer indicates the number of binary variables. We will assume that the set is bounded and has the following nonlinear description:
In descriptions (1)–(4) above the set contains the indices of the binary variables describing ; for each index belonging to the vectors and belong to , where is a positive integer indicating the number of constraints in (1). The th component of the two vectors and will be denoted by and , respectively. Finally, defined by the constraints (1), (2), and (3) will denote the continuous (or linear) relaxation of the mixed integer set .
In the sequel, two linear descriptions are said to be equivalent if they define the same polyhedron. A linear description dominates another linear description if the polyhedron defined by is included in the polyhedron defined by .
Optimizing even a linear function over the mixed integer set is an NP-hard problem in general (see [1–4]). A way of building strengthened linear relaxations is to use an approach combining reformulation, linearization, and projection such as those proposed in [5–10]. In such approach, one first reformulates the constraints defining the set of feasible solutions by introducing nonlinearities. Then, the resulting nonlinear system is linearized and projected back onto the original space.
Two important properties characterize the reformulation-linearization approach (also known as Lift-and-Project methods (do not confuse this with the Lift-and-Project hierarchy introduced by Balas et al., see [5])). First, the approach leads to a whole hierarchy (see [10, 11]) of relaxations which lie between the continuous relaxation and the convex hull of the mixed integer set . And for a given hierarchy, a relaxation of higher rank (see [10, 11]) is always stronger than a relaxation of lower rank. Secondly, optimizing a linear function over any relaxation of the hierarchy can be done in a polynomial time.
Many hierarchies were introduced. To mention a few, hierarchy (Balas et al., see [5]), hierarchy (Lovász and Schrijver, see [7]), hierarchy (Sherali and Admas, see [9, 10]), hierarchy (Lasserre, see [6]), hierarchy (Bienstock and Zuckerberg, see [12]), and hierarchy (Minoux and Ouzia, see [8, 13]). For more details, a set-theoretical interpretation of the reformulation-linearization approaches has been proposed in [14] and a theoretical comparative study between the , , and relaxations can be found in [15].
The convexification technique is also widely used to solve nonlinear and nonconvex optimization problems (see [16] and the references therein). Roughly speaking, this technique consists in approximating a nonconvex optimization problem by a convex problem (or a family of convex problems). This can be done by approximating the nonconvex objective function by a convex function and/or by approximating the nonconvex set of feasible solutions by a convex one (see [17–24] and the references therein). In this paper, new hierarchies of continuous relaxations using a reformulation, convexification, and linearization approach will be defined. Theses hierarchies are obtained using two different convexification schemes.
The paper is organized as follows. In the second section, first, we will recall the definition of a reformulation-linearization hierarchies and then give the definition of the Sherali-Adams hierarchy. In the third section, we will define two reformulation-convexification hierarchies: and hierarchies. We will study the main properties of these new hierarchies. In the fourth section, we will study the connections between , , and hierarchies. In the fifth section, we will introduce a weak version of the hierarchies and and emphasize some connections between them. In the last section we make some concluding remarks.
2. The Reformulation-Linearization Hierarchies
First, we will introduce the general concept of reformulation-linearization hierarchies. Then, we recall the definition of the well-known Sherali-Adams hierarchy (for more details see [9, 10, 25]).
Let be a positive integer. For a finite nonempty set let be the set of all subsets of with cardinality , whereas is the set of all subset of with cardinality at most . Sometimes we will need to indicate the cardinality of the sets under consideration, so we will use the notation to indicate (do not confuse this with the Cartesian product of sets) that the set has cardinality .
Let be a set of elements belonging to the set of binary indices and let be a subset from . We call -factor associated with the sets and , denoted by , the degree polynomial defined as follows: with the convention that .
Example 1. In the case where and we have the following nontrivial -factors: , , , , , , , , , , , and .
A rank reformulation-linearization relaxation (of the mixed integer set described by (1)–(4)) is defined in three steps. First, the problem is reformulated as a 0-1 polynomial (semialgebraic (a -dimensional semialgebraic set is a solution set of a finite system of polynomial equalities and inequalities; for more details see [26, 27])) mixed integer system by multiplying constraints (1)–(3) with all possible -factors (that is multiplying by for all subsets of and all ). Then, the nonlinear terms are linearized by replacing them with new variables giving rise to a higher dimensional linear system. The third step consists in projecting back the resulting polyhedron onto the original -space. As observed in [8] the linearization step can be performed in various ways, leading to various hierarchies of relaxations.
The solution set in associated with the nonlinear (semialgebraic) description resulting from the reformulation step will be denoted by and it is defined as follows:where, for each subset of , is the solution set defined by the following nonlinear system:
Starting from this semialgebraic reformulation, various linear relaxations can be constructed depending on the type of linearization considered (for more details see [8, 13]).
2.1. The Sherali-Adams Hierarchy
The description of the rank Sherali-Adams relaxation for the mixed integer set defined by (1)–(4), denoted by , is a reformulation-linearization relaxation of rank where the nonlinear terms appearing in (7) are linearized by introducing a new set of variables and defined bywhere it is assumed that and for every index (belonging to ) of a continuous variable.
Example 2. In the case where and we have the following linearization:
The resulting higher dimensional linear description will be denoted by and it is defined as follows:where, for each cardinality subset of , the linear description of the polyhedron isand where, for every index belonging to , and denote the linearized forms of the polynomials and , respectively; these are related to the and variables as follows:The above relations (12) are easily obtained by expanding the products involved in the definition of the -factors.
After linearizing the nonlinear terms in (7) using the variables defined in (8) above, the description turns out to involve a number of variables and constraints exponential in . The number of variables needed to linearize the nonlinear system (7) is (notice that the variable is not counted here since ). Also, it is seen that the number of constraints is .
The rank Sherali-Adams relaxation is obtained by projecting the polyhedron onto the subspace of the variables.
3. Two New Reformulation-Convexification Hierarchies
We will consider two new reformulation-convexification hierarchies. The first one is called reformulation-term-convexification () hierarchy. It is obtained by convexifying the monomials (also called terms) resulting from the reformulation step. The second hierarchy is called reformulation-standard-convexification (). It is obtained by convexifying the nonlinear factors (a linear combination of monomial products) resulting from the reformulation step.
3.1. Reformulation-Term-Convexification Hierarchy
A rank relaxation of the hierarchy is obtained by applying local convexification to each constraint of the nonlinear system defining as follows. For every subset from and every subset from with at most one element, let be the following operator:with the convention that and, for any real , is equal to .
The convexification scheme (13) assumes that the constraint to which it is applied is of the form ≤.
Let be the following convex set:where, for each cardinality subset from , the convex set corresponds to the solution set defined by the nonlinear system deduced from (7) by convexification using scheme (13). The convex nonlinear description of the set , for a given , readswhere, for every index belonging to and any scalar , and denote the convexified forms of the polynomials and , respectively; these are defined using the operator as follows:
Since the set is the intersection of convex sets (by construction), then we have the following result.
Theorem 3. For every integer belonging to , the set as defined by (15) is a nonlinear convex relaxation of the mixed integer set .
For every integer , let be the set (the lower-script is used to recall that our set is related to the term convexification scheme):where, for subsets and ,
Since every binary vector from belongs to then we deduce that coincides with the hypercube .
Let be the extended linear description obtained from the set using the following steps. Let and be two sets of additional variables such that for every -element set , for every subset from and for every subset from with at most one element. First, in (15), the variable will replace the nonlinear term:and the variable will replaceThen, we impose the equality constraint:We will use the notation or instead of when is empty or when coincides with the singleton , respectively.
Thus, we havewhere, for each subset of , the polyhedron readsand where, for every index belonging to , and denote the linearized forms of the convexified form of the polynomials and , respectively; these are related to the and variables as follows:The linear description (23)–(31) and (32) are stated using only the variables . This is possible according to (21). As discussed in Section 5, discarding constraints (21) in the definition of the hierarchy will lead to a weaker hierarchy.
For every integer belonging to , let be the projection onto the -space of the extended linear description . The polyhedron will be called rank- reformulation-term-convexification relaxation of .
In the next theorem we will prove that the hierarchy is equivalent to the hierarchy .
Theorem 4. For every integer belonging to , the two linear relaxations and are equivalent.
Proof. We will proceed by showing that the two extended linear descriptions and are the same up to variable renaming. Let be an integer belonging to . As shown in [8], for every set belonging to , the constraintsare implicit in the linear description of . We claim that the constraintsare also implicit in the linear description of . The argument is obvious for constraints (35). For constraints (36), let be an index belonging to and let be a subset from , and the following constraint is valid for . Using linearization, we get the constraint Combining this last constraint with constraint (34) we deduce that Now consider the following identifications: These identifications imply identifications between the and variables through (12) and (32). Thus, the two extended linear descriptions and are equivalent. This completes the proof.
Consequently, the rank- relaxation coincides with the convex hull of the mixed integer set . The hierarchy is motivated by the next theorem where it is shown that the projection onto the -space of any relaxation can be sandwiched between two convex sets. The following proposition will be useful.
Proposition 5. Let and be two disjoint subsets from , such that belongs to for all belonging to ; then
Theorem 6. For every integer belonging to , one has
Proof. First, to prove the left inclusion in (42) it is sufficient to prove that the set is included in . Let be a point belonging to the set . Let be a vector such thatThus, by definition constraints (31) are fulfilled by the vector . Since belongs to , then satisfies constraints (23)–(26). By Proposition 5 the point satisfies constraints (27)–(30). Since the vector also belongs to the set , then the vector also satisfies constraints (21). Consequently, the vector belongs to . Thus, belongs to .
Now, let us prove the right inclusion in (42). Let be a point belonging to . There is a vector such that belongs to . Without loss of generality, any constraint defining can be written as follows:where , and are subsets of the power set of (the superscript of a coefficient indicates its sign). Since the point satisfies the following inequalities: then we deduceConsequently, the point satisfies all constraints defining : that is, . This completes the proof.
Characterization (42) will allow us, in some cases (see Corollary 8), to give an explicit characterization of any relaxation (characterization in the -space). Before answering this question we will recall, in the next proposition, the simple result stating that any fractional point belonging to the set will belong to either a facet or an edge of the hypercube. This implies that the set has empty interior.
Proposition 7. For avery subset from , such that for all belonging to the set , ifthen at least variables are equal to .
As a consequence of Theorem 6 and Proposition 7 we have the following corollary.
Corollary 8. For every integer belonging to , if the set has integer vertices then it coincides with the projection of onto the -space.
Proof. Let be an integer belonging to . On the one hand, the sets and are both subsets from the hypercube. On the other hand, if the set has integer vertices then both sets and have the same vertices (vertices of ). Thus, the convex envelope of the set coincides with the set , because is convex. Consequently, using Theorems 4 and 6 we conclude that This completes the proof.
The following example shows that restricting the set to have integer vertices in Corollary 8 is not a sufficient condition to characterize . Let us consider the following set:Its continuous relaxation is the shaded region drawn in Figure 1. The set associated with the set is the shaded region drawn in Figure 2. A careful analysis of the set shows that it features the following linear description:
The rank-1 relaxation has the same linear description as . But, as shown in Figure 2, the set has a fractional vertex. Notice that set (49) coincides with the the rank-1 relaxation. As discussed in Section 5, this equality is not true in general. Finally, the set (the shaded region in Figure 3) has integer vertices and it coincides with the rank- relaxation.
3.2. Reformulation-Standard-Convexification Hierarchy
Contrary to the hierarchy, in the reformulation-standard-convexification hierarchy () we convexify each factor obtained after reformulation and not the monomials appearing in each such factors. The term standard extension was introduced by Crama (see [24]) in studying concave envelopes of pseudo-boolean functions.
Let be the nonlinear convex set:where, for each cardinality subset of , the convex set corresponds to the solution set of defined by the nonlinear system deduced from (7) using the following convexification scheme: where and are two subsets such that , , and .
As for the operator, depending on the type of the constraint ( or ), we use the expression or the expression in such a way that the resulting solution set will be convex.
The nonlinear description of the set , for a given and a set , is thus defined as follows:where, for every index belonging to and any scalar , and denote the convexified forms of the polynomials and using the operator , respectively.
Since the set as defined in (51) is the intersection of convex sets then we have the following results.
Theorem 9. For every integer belonging to , the set as defined by (51) is a nonlinear convex relaxation of the mixed integer set .
For every integer , let be the set (the lower-script is used to recall that our set is related to the standard convexification scheme) where for subsets and the set is the subset from defined by the following constraints:
Since every binary vector from belongs to then the set coincides with the hypercube . Notice that the set is a subset from . As shown below, it is possible to represent as a polyhedron in some appropriate extended space.
Let denote the extended linear description of the set obtained by using the following steps. First, let and be two sets of variables such that, for every -element set , for every subset from , and for every subset from with at most one element, replaces in (53) the nonlinear term:and the variable replacesThen we impose the following equality:
In the sequel, instead of or we will use or when the set coincides with the singleton and or otherwise (recall that the set has at most one element).
Thus, we obtain the extended linear description:where, for each subset of , the polyhedron reads
Notice that linear description (60)–(69) are stated using only the variables . This is possible according to (58). As discussed in the last section, discarding constraints (58) in the definition of the hierarchy will lead to a weaker hierarchy.
For every integer belonging to , let be the projection onto the -space of the extended linear description . The continuous relaxation will be called rank- reformulation-standard-convexification relaxation of the mixed integer set .
As for the hierarchy, in the following theorem we will prove that any relaxation of the hierarchy can also be sandwiched between two convex sets.
Theorem 10. For every integer belonging to , we have
Proof. Let be an integer belonging to . First, to prove the left inclusion in (70) it is sufficient to prove that the set is a subset of . Let be a point belonging to the set . Let be a vector where the two vectors and are defined as follows: for every -element set , for every subset from , and for every subset from with at most one element:By definition, belongs to ; then satisfies constraints (60)–(63). By Proposition 5 the point also satisfies constraints (64)–(69). Since the vector also belongs to the set , then the vector also satisfies constraints (58). Thus, the vector belongs to . Consequently, belongs to . This completes the first part of the proof.
Now, let us show the right inclusion in (70). Let be a point belonging to . There is a vector such that belongs to . Without loss of generality, each constraint in (60) and (61) can be rewritten as follows:Without loss of generality, we can assume that is nonnegative (the argument we will use holds also in the case where is nonpositive). The point satisfies also the constraints: Thus, we have This means that the point satisfies constraints (53). Thus, belongs to . This completes the proof.
4. Links between , , and Extended Linear Descriptions
We will focus, in this section, on the connections between the extended linear descriptions of the , , and relaxations. First, we will compare the strength of the and hierarchies. We will prove that for every rank the relaxation dominates the relaxation .
Let be a set of indices and let be an integer less than or equal to . Let be the subset from defined as follows:
The following lemmas will be useful to prove the next theorem.
Lemma 11. Let be a subset of indices from . Let be the set , where does not belong to . Let us consider the variables satisfyingThen, we have the following equality:
Proof. Let be the set and let be the set , where does not belong to . Let us consider the variables satisfying (76)-(77). First, notice that the set is a partition of the power set of . Thus, Using (77) we obtain
Lemma 12. Let be an integer belonging to . Let be a -element subset from . Let us consider the nonnegative variables , where is a subset from , satisfyingIf the variables satisfy the following inequalities:then the variables satisfy
Proof. Let be an integer belonging to . Let be a finite set and let be a -element subset from . It is a well-known fact that the linear transformation (82) is a bijection (see [10]) and its inverse is given byFirst, the variables satisfy inequalities (85) because of (88) and (83) and the fact that the variables are all nonnegative. Then, using Lemma 11, which is legitimate because the variables are assumed nonnegative, we deduce the inequalities (86). Finally, notice that for every subset from we have Using inequalities (84) we deduce that which is equivalent toThus, for sets and the variable satisfies inequalities (87) and this completes the proof.
Theorem 13. For every integer belonging to we have
Proof. Let be an integer belonging to . We will prove that the extended linear description is contained in . Let be a point belonging to . Let us define the point as follows:Since the point satisfies constraints (23)–(25) and (32), then it also satisfies constraints (60)–(63). Particularly, the variables are nonnegative. Using Lemma 12, where the variables are replaced by the variables and the variables are replaced by the variables , we deduce that the point satisfies constraints (66), (67), and (69). That is,We use the same arguments to prove that the point satisfies constraints (64), (65), and (68): that is,Thus, by identifying with and with we conclude that the point satisfies all constraints (60)–(63). This completes the proof.
Now, we will compare the strength of the two hierarchies and . As shown in [8], the hierarchy can be obtained from the semialgebraic set (6) using a suitable linearization (for more details, see [8]). Precisely, any rank- relaxation is a rank- reformulation-linearization relaxation where the linearization is performed using the following substitutions: for every subset from , letThe linearized system we obtain readsLet be the extended linear description (97). As before, let be its projection onto the -space.
In the next theorem we will prove that the hierarchy dominates the hierarchy .
Theorem 14. For every integer belonging to the set we have
Proof. Let be an integer belonging to the set . Let be a point belonging to . There exists a variable such that belongs to . Notice that, for every -element subset in , every subset from , and for every index from we have Constraints (97) and both relations (99) and (100) imply that the point satisfies constraints (60)–(69). Thus, the point belongs to . Consequently, the point belongs to . This completes the proof.
As a byproduct of Theorem 14 we obtain an indirect proof of Theorem 13. Indeed, for any rank , we know from Theorem 4 that is equivalent to . It is a well-known fact that is included in (see [8]). Thus, it follows using Theorem 14 that is included in .
5. Weak and Weak Hierarchies
In this section, we will introduce a weak version of the and hierarchies. For both weak hierarchies the rank extended linear description is obtained by reformulation, convexification using min and max and then linearizing using two distinct sets of variables.
More precisely, the rank extended linear description of the weak- hierarchy, denoted by , is defined as except that we discard equality constraints (21) (see Section 3.1). Thus, both sets of variables and will appear in the description of . Similarly, the rank extended linear description of the weak- hierarchy, denoted by , is defined as except that we discard equality constraints (58) between the two sets of variables and (see Section 3.2).
To emphasize some connections (small instances are sufficient to reveal these connections. We wish to emphasize that this is not a computational investigation) between the weak hierarchies we will use the computational results shown in Table 1. The values computed are the minimum value of rank 1 , , , , and relaxations for five instances of the multiple constraints knapsack problem. Each instance has constraints and variables. The instances have been generated using the Chu and Beasley procedure given in [28]. The constraint matrix coefficients are integers and randomly chosen in . The right-hand-side coefficient of the th constraint is set to . The th objective function coefficient is set to , where is a real number randomly chosen from the interval .
Although is stronger than the and hierarchies are not comparable in strength as shown by the instances inst-1 and inst-2. The hierarchy may be stronger than hierarchy as shown by the instances inst-2 and inst-4. Also, the computational results shown in the Table 1 are coherent with the theoretical results proved before: (i) the hierarchy is stronger than both and hierarchies; (ii) the hierarchy is stronger than hierarchy ; (iii) the hierarchies and are stronger than and , respectively.
6. Conclusion
In this paper, we introduced two new hierarchies called and for which the rank continuous relaxations were denoted by and , respectively. These two hierarchies are obtained using a reformulation-convexification-linearization procedure. The hierarchy is obtained using a term convexification scheme and the hierarchy is obtained using a standard convexification scheme. Then we compared the strength of these two hierarchies. We proved that (i) the hierarchy is equivalent to the hierarchy of Sherali-Adams, (ii) the hierarchy dominates the hierarchy , and (iii) the hierarchy is dominated by the Lift-and-Project hierarchy. Next, for every rank , we proved that and , where the sets and are convex, while and are two nonconvex sets with empty interior. The first inclusions allow, in some cases, an explicit characterization of relaxations. That is a convex nonlinear description of any relaxation in the -space. Finally, we discussed weak version of both and hierarchies and emphasized some connections between them using small numerical examples.
We conclude with some open questions. First, one may ask whether the hierarchy is equivalent to the hierarchy or not. Also, does the rank relaxation coincide with the convex envelope of the set ? Finally, is it possible to obtain stronger hierarchies using the exposed reformulation-convexification-linearization approach? Extending this work to more general nonlinear optimization problems will be the subject of a future work.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author would like to thank the anonymous referees for their helpful comments.