First, we introduce two new reformulation convexification based hierarchies called RTC and RSC for which the rank d continuous relaxations are denoted by P^RTCd and P^RSCd, 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 d, we will prove that convTd∩Etd⊆P^RTCd⊆Td and convSd∩Esd⊆P^RSCd⊆Sd where the sets Td and Sd are convex, while Etd and Esd 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 n and m be two integers. Let E and N be the two sets 1,…,n and 1,…,n+m, respectively. Let X be a subset of R+n+m representing the set of feasible solutions of a mixed integer linear program. The integer n indicates the number of binary variables. We will assume that the set X is bounded and has the following nonlinear description:(1)∑j∈Najxj≤b,(2)xj≤1,j∈N,(3)-xj≤0,j∈N,(4)xj∈0,1,j∈E.
In descriptions (1)–(4) above the set E contains the indices of the n binary variables describing X; for each index j belonging to N the vectors aj and b belong to Rρ, where ρ is a positive integer indicating the number of constraints in (1). The ith component of the two vectors aj and b will be denoted by aij and bi, respectively. Finally, X¯ defined by the constraints (1), (2), and (3) will denote the continuous (or linear) relaxation of the mixed integer set X.
In the sequel, two linear descriptions are said to be equivalent if they define the same polyhedron. A linear description D1 dominates another linear description D2 if the polyhedron defined by D1 is included in the polyhedron defined by D2.
Optimizing even a linear function over the mixed integer set X 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 X¯ and the convex hull of the mixed integer set X. 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, L&P hierarchy (Balas et al., see [5]), LS hierarchy (Lovász and Schrijver, see [7]), RLT hierarchy (Sherali and Admas, see [9, 10]), LAS hierarchy (Lasserre, see [6]), BZ hierarchy (Bienstock and Zuckerberg, see [12]), and DRL∗ 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 RLT, LS, and LAS 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: RTC and RSC hierarchies. We will study the main properties of these new hierarchies. In the fourth section, we will study the connections between RTC, RSC, and L&P hierarchies. In the fifth section, we will introduce a weak version of the hierarchies RTC and RSC 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 p be a positive integer. For a finite nonempty set S let Sp be the set of all subsets of S with cardinality p, whereas Sp is the set of all subset of S with cardinality at most p. Sometimes we will need to indicate the cardinality of the sets under consideration, so we will use the notation Hp to indicate (do not confuse this with the Cartesian product of sets) that the set H has cardinality p.
Let Jd be a set of d elements belonging to the set of binary indices E and let J be a subset from Jd. We call d-factor associated with the sets J and Jd, denoted by FdJ,Jd∖J, the degree d polynomial defined as follows:(5)FdJ,Jd∖J=∏j∈Jxj∏j∈Jd∖J1-xj, with the convention that ∏j∈∅yj=1.
Example 1.
In the case where E=1,2,3 and d=2 we have the following 12 nontrivial 2-factors: x1x2, x1x3, x2x3, 1-x1x2, 1-x1x3, 1-x2x1, 1-x2x3, 1-x3x1, 1-x3x2, 1-x11-x2, 1-x11-x3, and 1-x21-x3.
A rank d reformulation-linearization relaxation (of the mixed integer set X described by (1)–(4)) is defined in three steps. First, the problem is reformulated as a 0-1 polynomial (semialgebraic (a n-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 d-factors (that is multiplying by FdJ,Jd∖J for all subsets Jd of E and all J⊆Jd). 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 x-space. As observed in [8] the linearization step can be performed in various ways, leading to various hierarchies of relaxations.
The solution set in Rn+m associated with the nonlinear (semialgebraic) description resulting from the reformulation step will be denoted by Rd and it is defined as follows:(6)Rd=⋂Jd∈EdRdJd,where, for each subset Jd of E, RdJd is the solution set defined by the following nonlinear system:(7)∑j∈NajxjFdJ,Jd∖J≤bFdJ,Jd∖J,J⊆Jd,xjFdJ,Jd∖J≤FdJ,Jd∖J,j∈N,J⊆Jd,-xjFdJ,Jd∖J≤0,j∈N,J⊆Jd,-FdJ,Jd∖J≤0,J⊆Jd.
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 d Sherali-Adams relaxation for the mixed integer set X defined by (1)–(4), denoted by P^RLTd, is a reformulation-linearization relaxation of rank d where the nonlinear terms appearing in (7) are linearized by introducing a new set of variables wJ and wJk defined by(8)wJ=∏j∈Jxj∀J⊆E with J≤mind+1,n,wJk=xk∏j∈Jxj∀k∈N∖E,J⊆E with J≤mind,n,where it is assumed that w∅=1 and w∅k=xk for every index k (belonging to N∖E) of a continuous variable.
Example 2.
In the case where E=1,2,3 and N=E∪4 we have the following linearization: (9)w1,2=x1x2,w1,3=x1x3,w2,3=x2x3,w14=x1x4,w24=x2x4,w34=x3x4.
The resulting higher dimensional linear description will be denoted by PRLTd and it is defined as follows:(10)PRLTd=⋂Jd∈EdPRLTdJd,where, for each cardinality d subset Jd of E, the linear description of the polyhedron PRLTdJd is(11)∑j∈NajWjJ,Jd-bW0J,Jd≤0,J⊆Jd,WjJ,Jd-W0J,Jd≤0,j∈N,J⊆Jd,WjJ,Jd≥0,j∈N,J⊆Jd,W0J,Jd≥0,J⊆Jd,and where, for every index j belonging to N∖E, WjJ,Jd and W0J,Jd denote the linearized forms of the polynomials xjFdJ,Jd∖J and FdJ,Jd∖J, respectively; these are related to the wJ and wJk variables as follows:(12)WjJ,Jd=∑J⊆H⊆Jd-1H∖JwH∪j,Jd,J⊆Jd,j∈E,WjJ,Jd=∑J⊆H⊆Jd-1H∖JwHj,Jd,J⊆Jd,j∈N∖E,W0J,Jd=∑J⊆H⊆Jd-1H∖JwH,Jd,J⊆Jd.The above relations (12) are easily obtained by expanding the products involved in the definition of the d-factors.
After linearizing the nonlinear terms in (7) using the w variables defined in (8) above, the PRLTd description turns out to involve a number of variables and constraints exponential in d. The number of variables needed to linearize the nonlinear system (7) is ∑k=1mind+1,nnk+m∑k=0dnk (notice that the variable w∅ is not counted here since w∅=1). Also, it is seen that the number of constraints is O(nd2dn+m).
The rank d Sherali-Adams relaxation P^RLTd is obtained by projecting the polyhedron PRLTd onto the subspace Rn+m of the x variables.
3. Two New Reformulation-Convexification Hierarchies
We will consider two new reformulation-convexification hierarchies. The first one is called reformulation-term-convexification (RTC) hierarchy. It is obtained by convexifying the monomials (also called terms) resulting from the reformulation step. The second hierarchy is called reformulation-standard-convexification (RSC). 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 d relaxation of the RTC hierarchy is obtained by applying local convexification to each constraint of the nonlinear system defining Rd as follows. For every subset J from E and every subset H from N∖E with at most one element, let l be the following operator:(13)lα∏k∈Hxk∏j∈Jxj=α∑j∈J∪Hxj-J∪H+1+ifα>0,αminxj:j∈J∪Hifα≤0,with the convention that ∏k∈∅xk=1 and, for any real r, r+ is equal to max0,r.
The convexification scheme (13) assumes that the constraint to which it is applied is of the form ≤.
Let Td be the following convex set:(14)Td=⋂Jd∈EdTdJd,where, for each cardinality d subset Jd from E, the convex set TdJd corresponds to the solution set RdJd defined by the nonlinear system deduced from (7) by convexification using scheme (13). The convex nonlinear description of the set TdJd, for a given d, reads(15)∑j∈NLjJ,Jdaij+L0J,Jd-bi≤0,J⊆Jd,i=1,…,ρ,LjJ,Jd1+L0J,Jd-1≤0,j∈N,J⊆Jd,LjJ,Jd1≥0,j∈N,J⊆Jd,L0J,Jd1≥0,J⊆Jd,where, for every index j belonging to N and any scalar σ, LjJ,Jdσ and L0J,Jdσ denote the convexified forms of the polynomials σxjFdJ,Jd∖J and σFdJ,Jd∖J, respectively; these are defined using the l operator as follows: (16)LjJ,Jdσ=∑J⊆H⊆Jdlσ×-1H∖J∏k∈H∪jxk,Jd,J⊆Jd,j∈E,LjJ,Jdσ=∑J⊆H⊆Jdlσ×-1H∖Jxj∏k∈Hxk,Jd,J⊆Jd,j∈N∖E,L0J,Jdσ=∑J⊆H⊆Jdlσ×-1H∖J∏k∈Hxk,Jd,J⊆Jd.
Since the set Td is the intersection of convex sets (by construction), then we have the following result.
Theorem 3.
For every integer d belonging to {1,…,n}, the set Td as defined by (15) is a nonlinear convex relaxation of the mixed integer set X.
For every integer d, let Etd be the set (the lower-script t is used to recall that our set is related to the term convexification scheme):(17)Etd=⋂H∈Ed,K∈N∖H1EtH∪K,where, for subsets H∈Ed and K∈N∖H1, (18)EtH∪K=x∈0,1n+m:∑j∈H∪Kxj-H∪K+1+=minxj:j∈H∪K.
Since every binary vector from 0,1n+m belongs to Etd then we deduce that convEtd coincides with the hypercube 0,1n+m.
Let PRTCd be the extended linear description obtained from the set Td∩Etd using the following steps. Let t and v be two sets of additional variables such that for every d-element set Jd, for every subset J from Jd and for every subset H from N∖Jd with at most one element. First, in (15), the variable tJH will replace the nonlinear term:(19)minxj:j∈J∪H,and the variable vJH will replace(20)∑j∈J∪Hxj-J∪H+1+.Then, we impose the equality constraint:(21)tJH=vJH.We will use the notation tJ or tJk instead of tJH when H is empty or when H coincides with the singleton k, respectively.
Thus, we have(22)PRTCd=⋂Jd∈EdPRTCdJd,where, for each subset Jd of E, the polyhedron PRTCdJd reads(23)∑j∈NajTjJ,Jd-bT0J,Jd≤0,J⊆Jd,(24)TjJ,Jd-T0J,Jd≤0,j∈N,J⊆Jd,(25)-TjJ,Jd≤0,j∈N,J⊆Jd,(26)-T0J,Jd≤0,J⊆Jd,(27)minxj:j∈J≥tJ,J⊆Jd,(28)∑j∈Jxj-J+1≤tJ,J⊆Jd,(29)minxj:j∈J∪k≥tJk,J⊆Jd,k∈N∖E,(30)xk+∑j∈Jxj-J≤tJk,J⊆Jd,k∈N∖E,(31)tJ,tJk≥0,J⊆Jd,k∈N∖E,and where, for every index j belonging to N, TjJ,Jd and T0J,Jd denote the linearized forms of the convexified form of the polynomials xjFdJ,Jd∖J and FdJ,Jd∖J, respectively; these are related to the tJ and tJk variables as follows:(32)TjJ,Jd=∑J⊆H⊆Jd-1H∖JtH∪j,Jd,J⊆Jd,j∈E,TjJ,Jd=∑J⊆H⊆Jd-1H∖JtHj,Jd,J⊆Jd,j∈N∖E,T0J,Jd=∑J⊆H⊆Jd-1H∖JtH,Jd,J⊆Jd.The linear description (23)–(31) and (32) are stated using only the variables tJH. This is possible according to (21). As discussed in Section 5, discarding constraints (21) in the definition of the RTC hierarchy will lead to a weaker hierarchy.
For every integer d belonging to 1,…,n, let P^RTCd be the projection onto the x-space of the extended linear description PRTCd. The polyhedron P^RTCd will be called rank-d reformulation-term-convexification relaxation of X.
In the next theorem we will prove that the hierarchy RTC is equivalent to the hierarchy RLT.
Theorem 4.
For every integer d belonging to 1,…,n, the two linear relaxations P^RLTd and P^RTCd are equivalent.
Proof.
We will proceed by showing that the two extended linear descriptions PRLTd and PRTCd are the same up to variable renaming. Let d be an integer belonging to 1,…,n. As shown in [8], for every set Jd belonging to E[d], the constraints(33)minxj:j∈J≥wJ,J⊆Jd,(34)∑j∈Jxj-J+1≤wJ,J⊆Jd,are implicit in the linear description of PRLTd. We claim that the constraints(35)minxj:j∈J∪k≥wJk,J⊆Jd,k∈N∖E,(36)xk+∑j∈Jxj-J≤wJk,J⊆Jd,k∈N∖E,are also implicit in the linear description of PRLTd. The argument is obvious for constraints (35). For constraints (36), let k be an index belonging to N∖E and let J be a subset from Jd, and the following constraint(37)xk+∏j∈Jxj1-xk≤1 is valid for P^RLTd. Using RLT linearization, we get the constraint(38)xk+wJ-wJk≤1. Combining this last constraint with constraint (34) we deduce that(39)xk+∑j∈Jxj-J≤wJk,J⊆Jd,k∈N∖E. Now consider the following identifications: (40)wJ=tJ∀J⊆E,J≤mind+1,n,wJk=tJkfor every k∈N∖E,J⊆E,J≤mind,n.These identifications imply identifications between the W and T variables through (12) and (32). Thus, the two extended linear descriptions PRLTd and PRTCd are equivalent. This completes the proof.
Consequently, the rank-nRTC relaxation coincides with the convex hull of the mixed integer set X. The RTC hierarchy is motivated by the next theorem where it is shown that the projection onto the x-space of any RTC relaxation can be sandwiched between two convex sets. The following proposition will be useful.
Proposition 5.
Let J and H be two disjoint subsets from N, such that xj belongs to 0,1 for all j belonging to J∪H; then (41)∑j∈Jxj+∑j∈H1-xj-J∪H+1+≤minminh∈Jxh,minh∈H1-xh.
Theorem 6.
For every integer d belonging to 1,…,n, one has(42)convTd∩Etd⊆P^RTCd⊆Td.
Proof.
First, to prove the left inclusion in (42) it is sufficient to prove that the set Td∩Etd is included in P^RTCd. Let x^ be a point belonging to the set Td∩Etd. Let x^,t^,v^ be a vector such that(43)t^J=minx^j:j∈J,J⊂E,J≤mind+1,n,t^Jk=minx^j:j∈J∪k,k∈N∖E,J⊂Jd,Jd∈Ed,v^J=∑j∈Jx^j-J+1+,J⊂E,J≤mind+1,n,v^Jk=x^k+∑j∈Jx^j-J+2+,k∈N∖E,J⊂Jd,Jd∈Ed.Thus, by definition constraints (31) are fulfilled by the vector x^,t^,v^. Since x^ belongs to Td, then x^,t^,v^ satisfies constraints (23)–(26). By Proposition 5 the point x^,t^,v^ satisfies constraints (27)–(30). Since the vector x^ also belongs to the set Etd, then the vector x^,t^,v^ also satisfies constraints (21). Consequently, the vector x^,t^,v^ belongs to PRTCd. Thus, x^ belongs to P^RTCd.
Now, let us prove the right inclusion in (42). Let x^ be a point belonging to P^RTCd. There is a vector t^ such that x^,t^ belongs to PRTCd. Without loss of generality, any constraint defining PRTCd can be written as follows:(44)∑J∈S1+aJ+tJ+∑J∈S2+,k∈N∖EbJ+tJk+∑J∈S1-aJ-tJ+∑J∈S2-,k∈N∖EbJ-tJk≤0,where S1+,S2+,S1-, and S2- are subsets of the power set of N (the superscript of a coefficient indicates its sign). Since the point x^,t^ satisfies the following inequalities: (45)∑j∈Jx^j-J+1+≤t^J≤minx^j:j∈J,J⊆Jd,x^k+∑j∈Jx^j-J+≤t^Jk≤minx^j:j∈J∪k,J⊆Jd,k∈N∖E,then we deduce(46)∑J∈S1+aJ+∑j∈Jx^j-J+1++∑J∈S2+,k∈N∖EbJ+x^k+∑j∈Jx^j-J++∑J∈S1-aJ-minj∈Jx^j+∑J∈S2-,k∈N∖EbJ-minj∈J∪kx^j≤0.Consequently, the point x^ satisfies all constraints defining Td: that is, P^RTCd⊆Td. This completes the proof.
Characterization (42) will allow us, in some cases (see Corollary 8), to give an explicit characterization of any RLT relaxation (characterization in the x-space). Before answering this question we will recall, in the next proposition, the simple result stating that any fractional point belonging to the set Et will belong to either a facet or an edge of the hypercube. This implies that the set Et has empty interior.
Proposition 7.
For avery subset J from N, such that 0<xj≤1 for all j belonging to the set J, if(47)∑j∈Jxj-J+1+=minxj:j∈J,then at least J-1 variables are equal to 1.
As a consequence of Theorem 6 and Proposition 7 we have the following corollary.
Corollary 8.
For every integer d belonging to 1,…,n, if the set Td has integer vertices then it coincides with the projection of PRLTd onto the x-space.
Proof.
Let d be an integer belonging to 1,…,n. On the one hand, the sets Td and Etd are both subsets from the hypercube. On the other hand, if the set Td has integer vertices then both sets Td∩Etd and Td have the same vertices (vertices of Td). Thus, the convex envelope of the set Td∩Etd coincides with the set Td, because Td is convex. Consequently, using Theorems 4 and 6 we conclude that (48)P^RLTd=Td.This completes the proof.
The following example shows that restricting the set Td to have integer vertices in Corollary 8 is not a sufficient condition to characterize P^RLTd. Let us consider the following set:(49)X=x,y∈0,12:-9x+6y≤1,-12x+15y≤6.Its continuous relaxation X¯ is the shaded region drawn in Figure 1. The set T1 associated with the set X is the shaded region drawn in Figure 2. A careful analysis of the set T1 shows that it features the following linear description: (50)T1=x,y∈0,12:-5x+6y≤1,-4x+3y≤0.
The set X¯.
The set T1.
The rank-1 RLT relaxation has the same linear description as T1. But, as shown in Figure 2, the set T1 has a fractional vertex. Notice that set (49) coincides with the the rank-1 L&P relaxation. As discussed in Section 5, this equality is not true in general. Finally, the set T2 (the shaded region in Figure 3) has integer vertices and it coincides with the rank-2RLT relaxation.
Contrary to the RTC hierarchy, in the reformulation-standard-convexification hierarchy (RSC) 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 Sd be the nonlinear convex set:(51)Sd=⋂Jd∈EdSdJd,where, for each cardinality d subset Jd of E, the convex set SdJd corresponds to the solution set of RdJd defined by the nonlinear system deduced from (7) using the following convexification scheme: (52)Lα∏k∈Hxk∏j∈K1-xj=α∑j∈Hxj+∑j∈K1-xj-K∪H+1+ifα>0,αminminj∈Hxj,mink∈K1-xkifα≤0,where H and K are two subsets such that K⊂E, H⊂N, and H∩N∖E≤1.
As for the l operator, depending on the type of the constraint (≥ or ≤), we use the min expression or the max expression in such a way that the resulting solution set will be convex.
The nonlinear description of the set SdJd, for a given d and a set Jd, is thus defined as follows:(53)∑j∈NLjJ,Jdaij+L0J,Jd-bi≤0,J⊆Jd,i=1,…,ρ,LjJ,Jd1+L0J,Jd-1≤0,j∈E,J⊆Jd,LjJ,Jd-1≤0,j∈N,J⊆Jd,L0J,Jd-1≤0,J⊆Jd,where, for every index j belonging to N and any scalar σ, LjJ,Jdσ and L0J,Jdσ denote the convexified forms of the polynomials σxjFdJ,Jd∖J and σFdJ,Jd∖J using the operator L, respectively.
Since the set Sd as defined in (51) is the intersection of convex sets then we have the following results.
Theorem 9.
For every integer d belonging to {1,…,n}, the set Sd as defined by (51) is a nonlinear convex relaxation of the mixed integer set X.
For every integer d, let Esd be the set (the lower-script s is used to recall that our set is related to the standard convexification scheme)(54)Esd=⋂Jd∈Ed,K∈N∖Jd1EsJd,K, where for subsets Jd∈Ed and K∈N∖Jd1 the set EsJd,K is the subset from 0,1n+m defined by the following constraints: (55)∑j∈J∪Kxj+∑j∈Jd∖J1-xj-d+1+=minminxj:j∈J∪K,min1-xj:j∈Jd∖J,J⊆Jd.
Since every binary vector from 0,1n+m belongs to Esd then the set convEsd coincides with the hypercube 0,1n+m. Notice that the set Esd is a subset from Etd. As shown below, it is possible to represent Sd∩Esd as a polyhedron in some appropriate extended space.
Let PRSCd denote the extended linear description of the set Sd∩Esd obtained by using the following steps. First, let G and D be two sets of variables such that, for every d-element set Jd, for every subset J from Jd, and for every subset K from N∖Jd with at most one element, GKJ,Jd replaces in (53) the nonlinear term:(56)minminh∈J∪Kxh,mink∈Jd∖J1-xk,and the variable DKJ,Jd replaces(57)∑h∈J∪Kxh+∑k∈Jd∖J1-xk-d+1+.Then we impose the following equality:(58)GKJ,Jd=DKJ,Jd.
In the sequel, instead of GKJ,Jd or DKJ,Jd we will use GjJ,Jd or DjJ,Jd when the set K coincides with the singleton j and G0J,Jd or D0J,Jd otherwise (recall that the set K has at most one element).
Thus, we obtain the extended linear description:(59)PRSCd=⋂Jd∈EdPRSCdJd,where, for each subset Jd of E, the polyhedron PRSCdJd reads(60)∑j∈NajGjJ,Jd-bG0J,Jd≤0,J⊆Jd,(61)GjJ,Jd-G0J,Jd≤0,j∈E,J⊆Jd,(62)-GjJ,Jd≤0,j∈N,J⊆Jd,(63)-G0J,Jd≤0,J⊆Jd,(64)minxk:k∈J∪j≥GjJ,Jd,j∈N∖Jd,J⊆Jd,(65)min1-xk:k∈Jd∖J≥GjJ,Jd,j∈N∖Jd,J⊆Jd,(66)minxk:k∈J≥G0J,Jd,J⊆Jd,(67)min1-xk:k∈Jd∖J≥G0J,Jd,J⊆Jd,(68)∑h∈J∪jxh+∑k∈Jd∖J1-xk-d+1≤GjJ,Jd,j∈N∖Jd,J⊆Jd,(69)∑h∈Jxh+∑k∈Jd∖J1-xk-d+1≤G0J,Jd,J⊆Jd.
Notice that linear description (60)–(69) are stated using only the variables G. This is possible according to (58). As discussed in the last section, discarding constraints (58) in the definition of the RSC hierarchy will lead to a weaker hierarchy.
For every integer d belonging to 1,…,n, let P^RSCd be the projection onto the x-space of the extended linear description PRSCd. The continuous relaxation P^RSCd will be called rank-d reformulation-standard-convexification relaxation of the mixed integer set X.
As for the RTC hierarchy, in the following theorem we will prove that any relaxation of the RSC hierarchy can also be sandwiched between two convex sets.
Let d be an integer belonging to 1,…,n. First, to prove the left inclusion in (70) it is sufficient to prove that the set Sd∩Esd is a subset of P^RSCd. Let x^ be a point belonging to the set Sd∩Esd. Let x^,G^,D^ be a vector where the two vectors G^ and D^ are defined as follows: for every d-element set Jd, for every subset J from Jd, and for every subset K from N∖Jd with at most one element:(71)G^KJ,Jd=minminh∈J∪Kx^h,mink∈Jd∖J1-x^k,D^KJ,Jd=∑h∈J∪Kx^h+∑k∈Jd∖J1-x^k-d+1+.By definition, x^ belongs to Sd; then x^,G^,D^ satisfies constraints (60)–(63). By Proposition 5 the point x^,G^,D^ also satisfies constraints (64)–(69). Since the vector x^ also belongs to the set Esd, then the vector x^,G^,D^ also satisfies constraints (58). Thus, the vector x^,G^,D^ belongs to PRSCd. Consequently, x^ belongs to P^RSCd. This completes the first part of the proof.
Now, let us show the right inclusion in (70). Let x^ be a point belonging to P^RSCd. There is a vector G^ such that x^,G^ belongs to PRSCd. Without loss of generality, each constraint in (60) and (61) can be rewritten as follows:(72)∑j∈N:αij>0αijGjJ,Jd+∑j∈N:αij<0αijGjJ,Jd-βiG0J,Jd≤0,J⊆Jd.Without loss of generality, we can assume that βi is nonnegative (the argument we will use holds also in the case where βi is nonpositive). The point x^,G^ satisfies also the constraints: (73)minx^k:k∈J≥G^0J,Jd,J⊆Jd,min1-x^k:k∈Jd∖J≥G^0J,Jd,J⊆Jd,∑h∈Jx^h+∑k∈Jd∖J1-x^k-d+1+≤G^0J,Jd,J⊆Jd,minx^k:k∈J∪j≥G^jJ,Jd,j∈N∖Jd,J⊆Jd,min1-x^k:k∈Jd∖J≥G^jJ,Jd,j∈N∖Jd,J⊆Jd,∑h∈J∪jx^h+∑k∈Jd∖J1-x^k-d+1+≤G^jJ,Jd,j∈N∖Jd,J⊆Jd.Thus, we have (74)∑j∈N:αij>0αij∑h∈Jx^h+∑k∈Jd∖J1-x^k-d+1++∑j∈N:αij<0αijminmink∈J∪jx^k,mink∈Jd∖J1-x^k-βiminmink∈Jx^k,mink∈Jd∖J1-x^kG0J,Jd≤∑j∈N:αij>0αijGjJ,Jd+∑j∈N:αij<0αijGjJ,Jd-βiG0J,Jd,J⊆Jd. This means that the point x^ satisfies constraints (53). Thus, x^ belongs to Sd. This completes the proof.
We will focus, in this section, on the connections between the extended linear descriptions of the RTC, RSC, and L&P relaxations. First, we will compare the strength of the RTC and RSC hierarchies. We will prove that for every rank d the relaxation P^RTCd dominates the relaxation P^RSCd.
Let I=i1,…,ip be a set of indices and let j be an integer less than or equal to I. Let Ij be the subset from I defined as follows:(75)Ij=∅,ifj=0,ik∈I:k≤j,otherwise.
The following lemmas will be useful to prove the next theorem.
Lemma 11.
Let I be i1,…,ip a subset of indices from E. Let K be the set i0∪I, where i0 does not belong to I. Let us consider the variables y satisfying(76)y∅,K=∑H⊆K-1HxH,(77)yik,K∖Ik-1=-∑ik⊆H⊆K∖Ik-1-1HxH,k∈1,…,p.Then, we have the following equality:(78)y∅,i0∪I=1-xi0-∑k=1pyik,K∖Ik-1.
Proof.
Let I be the set i1,…,ip and let K be the set i0∪I, where i0 does not belong to I. Let us consider the variables y satisfying (76)-(77). First, notice that the set(79)H⊆K:H⊆i0∪⋃k=1pH⊆K:ik⊆H⊆K∖Ik-1 is a partition of the power set of K. Thus,(80)y∅,K=∑H⊆K-1HxH,=∑H⊆i0-1HxH+∑k=1p∑ik⊆H⊆K∖Ik-1-1HxH. Using (77) we obtain (81)y∅,K=∑H⊆i0-1HxH-∑k=1pyik,K∖Ik-1=1-xi0-∑k=1pyik,K∖Ik-1.
Lemma 12.
Let d be an integer belonging to 1,…,n. Let Jd be a d-element subset from E. Let us consider the nonnegative variables yJ,Jd, where J is a subset from Jd, satisfying(82)yJ,Jd=∑J⊆H⊆Jd-1H∖JxH,J⊆Jd.If the variables x satisfy the following inequalities:(83)xH≤minxj:j∈H,H⊆Jd,(84)xH≥∑k∈Hxk-H+1+,then the variables y satisfy(85)yJ,Jd≤minxj:j∈J,J⊆Jd,(86)yJ,Jd≤min1-xj:j∈Jd∖J,J⊆Jd,(87)yJ,Jd≥∑k∈Jxk-∑k∈Jd∖Jxk-J+1+.
Proof.
Let d be an integer belonging to 1,…,n. Let E be a finite set and let Jd be a d-element subset from E. It is a well-known fact that the linear transformation (82) is a bijection (see [10]) and its inverse is given by(88)xJ=∑J⊆H⊆JdyH,Jd,J⊆Jd.First, the variables yJ,Jd satisfy inequalities (85) because of (88) and (83) and the fact that the variables yJ,Jd are all nonnegative. Then, using Lemma 11, which is legitimate because the variables yJ,Jd are assumed nonnegative, we deduce the inequalities (86). Finally, notice that for every subset J from Jd we have(89)xJ≤yJ,Jd+∑k∈Jd∖JxJ∪k. Using inequalities (84) we deduce that(90)yJ,Jd+∑k∈Jd∖JxJ∪k≥∑k∈Jxk-J+1, which is equivalent to(91)yJ,Jd≥∑j∈Jxj-∑k∈Jd∖JxJ∪k-J+1.Thus, for sets J and Jd the variable yJ,Jd satisfies inequalities (87) and this completes the proof.
Theorem 13.
For every integer d belonging to 1,…,n we have(92)P^RTCd⊆P^RSCd.
Proof.
Let d be an integer belonging to 1,…,n. We will prove that the extended linear description PRTCd is contained in PRSCd. Let x^,t^ be a point belonging to PRTCd. Let us define the point x^,G^ as follows:(93)G^jJ,Jd=∑J⊆H⊆Jd-1H∖Jt^H∪j,Jd,J⊆Jd,j∈E,G^jJ,Jd=∑J⊆H⊆Jd-1H∖Jt^Hj,Jd,J⊆Jd,j∈N∖E,G^0J,Jd=∑J⊆H⊆Jd-1H∖Jt^H,Jd,J⊆Jd.Since the point x^,t^ satisfies constraints (23)–(25) and (32), then it also satisfies constraints (60)–(63). Particularly, the variables G^ are nonnegative. Using Lemma 12, where the variables y are replaced by the variables G^ and the variables x are replaced by the variables t^, we deduce that the point x^,G^ satisfies constraints (66), (67), and (69). That is,(94)minx^k:k∈J≥G^0J,Jd,J⊆Jd,min1-x^k:k∈Jd∖J≥G^0J,Jd,J⊆Jd,∑h∈Jx^h+∑k∈Jd∖J1-x^k-d+1≤G^0J,Jd,J⊆Jd.We use the same arguments to prove that the point x^,G^ satisfies constraints (64), (65), and (68): that is,(95)minx^k:k∈J∪j≥G^jJ,Jd,j∈N∖Jd,J⊆Jd,min1-x^k:k∈Jd∖J≥G^jJ,Jd,j∈N∖Jd,J⊆Jd,∑h∈J∪jx^h+∑k∈Jd∖J1-x^k-d+1≤G^jJ,Jd,j∈N∖Jd,J⊆Jd.Thus, by identifying G^jJ,Jd with TjJ,Jd and G^0J,Jd with T0J,Jd we conclude that the point x^,G^ satisfies all constraints (60)–(63). This completes the proof.
Now, we will compare the strength of the two hierarchies RSC and L&P. As shown in [8], the L&P hierarchy can be obtained from the semialgebraic set (6) using a suitable linearization (for more details, see [8]). Precisely, any rank-dL&P relaxation is a rank-d reformulation-linearization relaxation where the linearization is performed using the following substitutions: for every subset Jd from E, let(96)ZjJ,Jd=xjFdJ,Jd∖J∀j∈N,J⊆Jd,Z0J,Jd=FdJ,Jd∖J∀J⊆Jd.The linearized system we obtain reads(97)∑j=1κajZjJ,Jd-bZ0J,Jd≤0∀J⊆Jd,Jd∈Ed,ZjJ,Jd-Z0J,Jd≤0∀j∈E,J⊆Jd,Jd∈Ed,ZjJ,Jd,Z0J,Jd≥0∀j∈N,J⊆Jd,Jd∈Ed.Let PL&Pd be the extended linear description (97). As before, let P^L&Pd be its projection onto the x-space.
In the next theorem we will prove that the hierarchy L&P dominates the hierarchy RSC.
Theorem 14.
For every integer d belonging to the set 1,…,n we have(98)P^L&Pd⊆P^RSCd.
Proof.
Let d be an integer belonging to the set 1,…,n. Let x^ be a point belonging to P^L&Pd. There exists a variable Z^ such that x^,Z^ belongs to PL&Pd. Notice that, for every d-element subset Jd in E, every subset J from Jd, and for every index j from N we have (99)∑k∈J∪jxk-∑k∈Jd∖Jxk-J+1+≤ZjJ,Jd≤minminh∈J∪jxh,mink∈Jd∖J1-xk,(100)∑k∈Jxk-∑k∈Jd∖Jxk-J+1+≤Z0J,Jd≤minminh∈Jxh,mink∈Jd∖J1-xk.Constraints (97) and both relations (99) and (100) imply that the point x^,Z^ satisfies constraints (60)–(69). Thus, the point x^,Z^ belongs to PRSCd. Consequently, the point x^ belongs to P^RSCd. This completes the proof.
As a byproduct of Theorem 14 we obtain an indirect proof of Theorem 13. Indeed, for any rank d, we know from Theorem 4 that P^RTCd is equivalent to P^RLTd. It is a well-known fact that P^RLTd is included in P^L&Pd (see [8]). Thus, it follows using Theorem 14 that P^RTCd is included in P^RSCd.
In this section, we will introduce a weak version of the RTC and RSC hierarchies. For both weak hierarchies the rank d 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 d extended linear description of the weak-RTC hierarchy, denoted by PWRTCd, is defined as PRTCd except that we discard equality constraints (21) (see Section 3.1). Thus, both sets of variables tJH and vJH will appear in the description of PWRTCd. Similarly, the rank d extended linear description of the weak-RSC hierarchy, denoted by PWRSCd, is defined as PRSCd except that we discard equality constraints (58) between the two sets of variables GKJ,Jd and DKJ,Jd (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 RTC, L&P, RSC, WRTC, and WRSC relaxations for five instances of the multiple constraints knapsack problem. Each instance has 5 constraints and 3 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 0,…,1000. The right-hand-side coefficient of the ith constraint is set to 0.95∑j=13aij. The jth objective function coefficient is set to 1/5∑i=15aij+500ζj, where ζj is a real number randomly chosen from the interval 0,1.
Optimal values of the different relaxations.
Instance
Optimum values
P^RTC1
P^L&P1
P^RSC1
P^WRTC1
P^WRSC1
inst-1
-2676.783
-2676.8066
-2677.0347
-3390.3195
-2677.0347
inst-2
-3390.3195
-3390.7008
-3391.2893
-3390.3195
-3391.2893
inst-3
-3010.0385
-3010.0385
-3010.4338
-3010.0385
-3010.4338
inst-4
-3229.0645
-3229.1293
-3229.1293
-3229.0645
-3229.1293
inst-5
-3337.7582
-3337.7582
-3337.8134
-3337.7582
-3337.8134
Although RTC is stronger than RSC the WRTC and WRSC hierarchies are not comparable in strength as shown by the instances inst-1 and inst-2. The WRTC hierarchy may be stronger than L&P 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 RTC is stronger than both L&P and RSC hierarchies; (ii) the hierarchy L&P is stronger than hierarchy RSC; (iii) the hierarchies RTC and RSC are stronger than WRTC and WRSC, respectively.
6. Conclusion
In this paper, we introduced two new hierarchies called RTC and RSC for which the rank d continuous relaxations were denoted by P^RTCd and P^RSCd, respectively. These two hierarchies are obtained using a reformulation-convexification-linearization procedure. The hierarchy RTC is obtained using a term convexification scheme and the RSC hierarchy is obtained using a standard convexification scheme. Then we compared the strength of these two hierarchies. We proved 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. Next, for every rank d, we proved that convTd∩Etd⊆P^RTCd⊆Td and convSd∩Esd⊆P^RSCd⊆Sd, where the sets Td and Sd are convex, while Etd and Esd are two nonconvex sets with empty interior. The first inclusions allow, in some cases, an explicit characterization of RLT relaxations. That is a convex nonlinear description of any RLT relaxation in the x-space. Finally, we discussed weak version of both RTC and RSC hierarchies and emphasized some connections between them using small numerical examples.
We conclude with some open questions. First, one may ask whether the hierarchy RSC is equivalent to the hierarchy WRSC or not. Also, does the rank nRSC relaxation coincide with the convex envelope of the set X? 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.
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