Convex separable minimization problems with a linear constraint and bounded variables

CONVEX SEPARABLE MINIMIZATION PROBLEMS WITH

A LINEAR CONSTRAINT AND BOUNDED VARIABLES

Received 30 June 2004 and in revised form 19 April 2005

Consider the minimization problem with a convex separable objective function over a feasible region defined by linear equality constraint(s)/linear inequality constraint of the form “greater than or equal to” and bounds on the variables. A necessary and sufficient condition and a sufficient condition are proved for a feasible solution to be an optimal so-lution to these two problems, respectively. Iterative algorithms of polynomial complexity for solving such problems are suggested and convergence of these algorithms is proved. Some convex functions, important for problems under consideration, as well as compu-tational results are presented.

1. Introduction

In many cases, we have to minimize a convex separable function over a region defined by a linear equality or inequality “≥” constraint with positive coefficients, and two-sided bounds on the variables.

Such problems and problems related to them arise, for example, in production plan-ning and scheduling [2] and Problem 4,Section 5, in allocation of resources [2,31] and Problem 1,Section 5, in allocation of effort resources among competing activities [16] and Problems 3, 5, and 6,Section 5, in the theory of search [6], in subgradient opti-mization [11], in facility location [24], and in the implementation of projection methods when the feasible region is of the considered form [28] and Problem 2,Section 5, and so forth.

The problems under consideration can mathematically be formulated as follows:

min

c(x)=

j∈J

cjxj

(1.1)

subject to

j∈J

djxj ₌

≥

α, (1.2)

aj≤xj≤bj, j∈J, (1.3)

Copyright©2005 Hindawi Publishing Corporation

wherecj(xj) are twice differentiable convex functions, defined on the open convex sets

XjinR,j∈J, respectively;dj>0 for every j∈J,x=(xj)j∈J, andJ≡ {1,...,n}.

Denote this problem by (C=) in the first case (problem (1.1), (1.2), and (1.3) with equality constraint (1.2)), and by (C≥) in the second case (problem (1.1), (1.2), and (1.3) with inequality “≥” constraint (1.2)). Denote byX=andX≥the feasible region (1.2)-(1.3) in the two cases, respectively. A constraint like (1.2) is known as theknapsack constraint.

Also, a generalization of problem (C=), denoted by (Cm=), is considered in which (1.2) is defined bymlinear equality constraints.

The feasible region (1.2)-(1.3) is an intersection of the hyperplane(s)/halfspace (1.2) and the box (1.3) of dimension|J| =n. Therefore, (1.2)-(1.3) is a convex set. Therefore, (C=), (C=_m), and (C≥) are convex programming problems.

Problems like (C=) and (C≥) are subject of intensive study. Related problems and methods for them are considered in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, 18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The solution of knapsack problems with arbitrary convex or concave objective functions is studied in [2,16,20,25,31], and so forth. Algorithms for solving convex quadratic minimization problems with a linear equality/inequality constraint and box constraints are proposed in [27]. Quadratic knap-sack problems and problems related to them are studied in [4,22,23], and so forth. A nonconvex variant of these problems is considered in [29], and algorithms for the case of convex quadratic objective function are proposed in [4,9,12,21], and so forth. Bounded knapsack sharing is considered in [3]. Algorithms for bound constrained quadratic pro-gramming problems are proposed in [19], and minimization of quadratic functions sub-ject to box constraints is considered in [8]. Quasi-Newton updates with bounds are sug-gested in [5]. A Lagrangian relaxation algorithm for the constrained matrix problem is proposed in [7]. Analytic solutions of nonlinear programs subject to one or two equality constraints are studied in [10], and minimization subject to one linear equality constraint and bounds on the variables is considered in [17]. Iterative quadratic optimization algo-rithms for pairs of inequalities are proposed in [13]. A polynomial-time algorithm for the resource allocation problem with a convex objective function and nonnegative inte-ger variables is suggested in [14]. Algorithms for the least-distance problem are proposed in [1,30]. Polynomial algorithms for projecting a point onto a region defined by a lin-ear constraint and box constraints inRn_{are suggested in [28]. An algorithm for finding} a projection onto a simple polytope is proposed in [18]. A method for solving a con-vex integer programming problem is suggested in [26]. The problems of maximizing and minimizing subsums subject to a sum- and a Schur-convex constraint are solved in [15] without the use of the theory of convex programming.

In this paper, we propose iterative algorithms (Sections3.2and3.4) which are based on Theorems2.1and2.4(Sections2.1and2.3, resp.). Convergence of these algorithms is based onTheorem 3.2(Section 3.3). Then we pay our attention to some extensions concerning theoretical and computational aspects of the proposed approach (Section 4). InSection 5, we give examples of convex functionscj(xj), which are involved in problems under consideration, and some computational results.

inequality constraint of the form “≤” and bounded variables, and generalization of the author’s paper [27], in which the special case ofquadraticseparable objective function is studied.

2. Main results: characterization theorems

2.1. Problem(C=). First consider the following problem (C=):

min

c(x)=

j∈J

cjxj

(2.1)

subject to

j∈J

djxj=α, dj>0, j∈J, (2.2)

aj≤xj≤bj, j∈J. (2.3)

Suppose that the following assumptions are satisfied.

(1a) aj≤bj for all j∈J. Ifak=bk for somek∈J, then the valuexk:=ak=bk is determined in advance.

(2a)_j∈Jdjaj≤α≤j∈Jdjbj. Otherwise, the constraints (2.2) and (2.3) are incon-sistent andX== ∅, whereX=is defined by (2.2) and (2.3).

Leth=_j, j∈J, be the value ofxj for whichcj(xj)=0. If a finite value with this prop-erty does not exist, this means that the functioncj(xj) does not change the type of its monotonicity, that is,cj(xj) is a nondecreasing or nonincreasing function in the interval (−∞, +∞). That is why, in case there does not exist a finite valueh=_j such thatc_j(h=_j)=0, we adopt the following:

(i)h=j := −∞, ifcj(xj) is a nondecreasing function; (ii)h=_j :=+∞, ifcj(xj) is a nonincreasing function. The Lagrangian for problem (C=) is

L(x,u,v,λ)=

j∈J

cj

xj

+λ

j∈J

djxj−α

+

j∈J

uj

aj−xj

+

j∈J

vj

xj−bj

, (2.4)

whereλ∈R1_,u,v∈Rn

+, andRn+consists of all vectors withnreal nonnegative compo-nents.

The Karush-Kuhn-Tucker (KKT) necessary and sufficient optimality conditions for the minimum solutionx∗=(x∗_j)j∈J for problem (C=) are

c_jx∗_j+λdj−uj+vj=0, j∈J, (2.5)

uj

aj−x∗j

=0, j∈J, (2.6)

vj

x∗j −bj

λ∈R1_{, u}

j∈R1+, vj∈R1+, j∈J, (2.8)

j∈J

djx∗j =α, (2.9)

aj≤x∗j ≤bj, j∈J, (2.10)

whereλ,uj,vj, j∈J, are the Lagrange multipliers associated with the constraints (2.2),

aj≤xj,xj≤bj,j∈J, respectively. Ifaj= −∞orbj=+∞for somej, we do not consider the corresponding condition (2.6) ((2.7), resp.) and Lagrange multiplieruj(vj, resp.).

Sincecj(xj), j∈J, are convex differentiable functions in one variable, thencj(xj) are monotone nondecreasing functions ofxj,j∈J, that is,

c_jx1 j

−c_jx2 j

x1 j−x2j

≥0 ∀x1

j,x2j, j∈J. (2.11) Sinceuj≥0,vj≥0, j∈J, and since the complementary conditions (2.6) and (2.7) must be satisfied, in order to findx∗_j, j∈J, from system (2.5), (2.6), (2.7), (2.8), (2.9), and (2.10), we have to consider all possible cases foruj,vj: alluj,vj equal to 0; alluj,

vj different from 0; some of them equal to 0 and some of them different from 0. The number of these cases is 2|J|₌₂2n_{, where 2nis the number of allu}

j,vj,j∈J, and|J| =

n. Obviously, this is an enormous number of cases, especially for large-scale problems. For example, whenn=1500, we have to consider 23000_≈10900_{cases. Moreover, in each} case, we have to solve a large-scale system of (nonlinear) equations inx∗_j,λ,uj,vj, j∈J. Therefore, thedirectapplication of the KKT theorem, using explicit enumeration of all possible cases, for solving large-scale problems of the considered form would not give a result and we need efficient methods to solve the problems under consideration.

Theorem 2.1gives necessary and sufficient condition (characterization) of the optimal solution to problem (C=). Its proof, of course, is based on the KKT theorem. As we will see inSection 5, by usingTheorem 2.1, we can solve problem (C=) withn=1500 variables for about 0.0001 seconds on a personal computer.

Theorem2.1 (characterization of the optimal solution to problem (C=)). A feasible

sol-utionx∗=(x∗j)j∈J∈X=is an optimal solution to problem(C=)if and only if there exists a

λ∈R1_{such that}

x∗_j =aj, j∈Jaλ def

=

j∈J:λ≥ −c

j

aj

dj

, (2.12)

x∗_j =bj, j∈Jbλ def

=

j∈J:λ≤ −c

j

bj

dj

, (2.13)

x∗_j :λdj= −cj

x∗_j, j∈Jλdef₌

j∈J:−c

j

bj

dj ≤λ≤ −

cj

aj

dj

. (2.14)

Whencj(xj)are strictly convex, inequalities definingJλin (2.14) are strict.

(a) Ifx∗_j =aj, thenuj≥0 andvj=0 according to (2.6), (2.7), and (2.8). Therefore, (2.5) implies thatcj(x∗j)=uj−λdj≥ −λdj. Sincedj>0, then

λ≥ −c

j

x∗_j

dj ≡ −

c_jaj

dj . (2.15)

(b) Ifx∗j =bj, thenuj=0 andvj≥0 according to (2.6), (2.7), and (2.8). Therefore, (2.5) implies thatc_j(x∗_j)= −vj−λdj≤ −λdj. Hence,

λ≤ −c

j

x∗_j

dj ≡ −

c_jbj

dj . (2.16)

(c) Ifaj< x∗j < bj, thenuj=vj=0 according to (2.6) and (2.7). Therefore, (2.5) im-plies that c_j(x∗_j)= −λdj. Sincecj(xj), j∈J, are convex differentiable functions, then

c_j(xj) are nondecreasing functions, and sincebj> x∗j,x∗j > aj,j∈J, by the assumptions, it follows that

c_jbj≥cj

x∗_j≥c_jaj, j∈J. (2.17) Multiplying these inequalities by−1/djand using that

dj>0, λ= −

c_jx∗_j

dj , (2.18)

we obtain

−c

j

bj

dj ≤λ≤ −

c_jaj

dj . (2.19)

Whencj(xj), j∈J, arestrictly convex, thenaj< x∗j < bj implies thatcj(bj)> cj(x∗j)>

c_j(aj), j∈J, strictly. Then inequalities in (2.19) are strict.

To describe cases (a), (b), and (c), we introduce the index setsJλ

a,Jbλ,Jλ defined by (2.12), (2.13), and (2.14), respectively. It is obvious thatJλ

a∪Jbλ∪Jλ=J. The “necessity” part ofTheorem 2.1is proved.

Sufficiency. Conversely, letx∗=(x∗_j)j∈J∈X= and components ofx∗satisfy (2.12), (2.13), and (2.14). Set

λ= −c

j

x∗j

dj ; uj=vj=0 forj∈J λ_;

uj=cj

aj+λdj (≥0), vj=0 forj∈Jaλ;

uj=0, vj= −cj

bj

−λdj (≥0) for j∈Jbλ.

(2.20)

Whencj(xj), j∈J, arestrictlyconvex, this optimal solution is unique. The importance ofTheorem 2.1consists in the fact that it describes components of the optimal solution to problem (C=) only through the Lagrange multiplierλassociated with the equality constraint (2.2).

Since we do not know the optimal value ofλfromTheorem 2.1, we define an iterative process with respect to the Lagrange multiplierλand we prove the convergence of this process inSection 3.

Using thatdj>0,j∈J, from monotonicity ofcj(xj) and fromaj≤bj,j∈J, it follows thatubjdef= −cj(bj)/dj≤ −cj(aj)/djdef=laj, j∈J, for the expressions by means of which we define the setsJλ

a,Jbλ, andJλ.

The problem how to ensure a feasible solution to problem (C=) defined by (2.1), (2.2), and (2.3), which is an assumption forTheorem 2.1, is discussed inSection 3.3.

The question whetherxj,j∈Jλ, are always uniquely determined from (2.14) in [aj,bj] is important. In general, ifc(x)≡j∈Jcj(xj) is a strictly convex function, then problem (C=) has a unique optimal solution in the feasible regionX=defined by (2.2) and (2.3) in case (C=) has a feasible solution, that is,x∗_j,j∈Jλ_{, are uniquely determined from (2.14)} in this case. If the parametersaj,bj, and so forth of particular problems of type (C=) are generated in intervals where the functionscj(xj) are strictly convex, then the optimal solution to the corresponding problem (C=), if it has an optimal solution, is unique.

Ifc(x) is a convex function but not necessarily a strictly convex function, then, as it is known, any local minimum point ofc(x) is a global minimum point as well and the set of optimal solutions to a convex minimization problem is convex. Therefore, the optimal value of the objective function subject to (2.2) and (2.3) is the same for all optimal solu-tions to (C=) if it has more than one optimal solution. If, for example, (2.14) is a linear equation ofx∗_j, thenx∗_j,j∈Jλ_{, are uniquely determined from (2.14) in this case as well.}

2.2. Problem(C=m). Denote by (C=m) the problem

min

c(x)=

j∈J

cjxj

(2.21)

subject to

Dx=α, (2.22)

a≤x≤b, (2.23)

wherecj(xj) are differentiable strictly convex functions,j∈J,D=(di j)∈Rm×n,α∈Rm, a=(a1,...,an), andb=(b1,...,bn)∈Rn.

The feasible region (2.22)-(2.23) is an intersection ofmhyperplanes (2.22) and the box (2.23).

Denote byPc(D,α,a,b) the solution to problem (C=m). Sincec(x) is strictly convex as a sum of strictly convex functions, thenPc(D,α,a,b) is uniquely defined, that is, there is at most one minimum which is both local and global.

Denotey=[x]b

a, whereyj=min{max{xj,aj},bj}for eachj∈J. The KKT conditions forx∗∈Rn_{to be a local minimum of (C}=

m) are

Dx∗=α, (2.24)

a≤x∗, (2.25)

x∗≤b, (2.26)

cx∗+DT_{λ−u+v=0, (2.27)}

uj

aj−x∗j

=0, j∈J, (2.28)

vjx∗j −bj=0, j∈J, (2.29)

u≥0, (2.30)

v≥0, (2.31)

whereλ∈Rm_,u,v∈Rn

+are the Lagrange multipliers associated with (2.22) and the two inequalities of (2.23), respectively.

The mapc≡ ∇c:Rn_→Rn_{is strict monotone increasing sincecis a strictly convex} function. Therefore, (∇c)−1_:Rn_→Rn_{is well defined.}

Theorem2.2. Letc:Rn_{→Rbe separable, differentiable, and strictly convex. Then,}

Pc(D,α,a,b)=

(c)−1_−DT_tc(b)

c_(a): t∈Rm

, (2.32)

whereD,α,a,bare defined above.

Proof. Relation (2.32) is proved by two-way inclusion.

(i) Letx∗=Pc(D,α,a,b) for someα∈Rm. Then there existλ∈Rm,u,v∈Rn+ satis-fying the KKT conditions (2.24), (2.25), (2.26), (2.27), (2.28), (2.29), (2.30), and (2.31) together with thisx∗.

It follows from (2.27) that

DT_{λ= −cx}∗_{+u−v, (2.33)}

that is,

Dj,λ

= −c

j

x∗_j+uj−vj (2.34)

for eachj∈J. IfD_j,λ>−c

j(x∗j), thenuj> vj≥0, sox∗j =ajaccording to (2.28), that is,

Dj,λ

>−c

j

x∗j

implies thatx∗j =aj. (2.35)

Similarly, ifDj,λ<−cj(x∗j), thenvj> uj≥0, sox∗j =bjaccording to (2.29), that is,

Dj,λ

<−c

j

Sinceaj≤bj,j∈J, by assumption, we have three cases to consider.

Case 1. Dj,λ>−cj(aj). ThenDj,λ>−cj(x∗j) according to (2.25) and the mono-tonicity ofcj. Hence,x∗j =ajin accordance with (2.35).

Case 2. Dj,λ<−cj(bj). ThenDj,λ<−cj(x∗j) according to (2.26) and the mono-tonicity ofc_j. Hence,x∗_j =bjin accordance with (2.36).

Case 3. −c_j(bj)≤Dj,λ ≤ −cj(aj). IfDj,λ<−cj(x∗j), thenx∗j =bjaccording to (2.36). Therefore,Dj,λ ≥ −cj(x∗j) becauseDj,λ ≥ −cj(bj) by the assumption ofCase 3, a contradiction. Similarly, if we assume thatDj,λ>−cj(x∗j) strictly, this would imply thatx∗j =ajaccording to (2.35) andDj,λ ≤ −cj(x∗j), a contradiction.

ThenD_j,λ = −c

j(x∗j), so it follows thatx∗j =(cj)−1(−Dj,λ). In the three cases considered, we have

x∗j =

cj −1

−Dj,λ c

j(bj)

cj(aj). (2.37)

Hence,x∗=(c)−1_[−DT_λ]c(b) c(a), that is,

Pc(D,α,a,b)⊆

(c)−1_−DT_tc(b)

c_(a):t∈Rm

. (2.38)

(ii) Conversely, suppose thatx∗∈Rn_andx∗_=(c)−1_[−DT_t]c(b)

c_(a)for somet∈Rm. Set

α=D(c)−1_−DT_tc(b) c_(a),

λ=t, u=c(a) +DTt, v= −c(b)−DT_t.

(2.39)

We have to prove thatx∗,α,λ,u,vsatisfy the KKT conditions (2.24), (2.25), (2.26), (2.27), (2.28), (2.29), (2.30), and (2.31).

Obviously,x∗ and αsatisfy (2.24), x∗ satisfies (2.25) and (2.26) (these are (2.23)) according to the definition of [x]b

aand the monotonicity ofc.

In order to verify (2.27), (2.28), (2.29), (2.30), and (2.31), we consider each j∈J. There are three possible cases.

Case 1. Dj,t>−cj(aj). Then cj(aj) +Dj,t>0, and sinceaj ≤bj, then−cj(bj)−

Dj,t<0. Therefore,x∗j =aj,λ=t,uj=cj(aj) +Dj,t,vj=0.

Case 2. D_j,t<−c

j(bj). Then −cj(bj)− Dj,t>0, and sinceaj≤bj, thencj(aj) +

Dj,t<0. Therefore,x∗j =bj,λ=t,uj=0,vj= −cj(bj)− Dj,t.

Case 3. −c

j(bj)≤ Dj,t ≤ −cj(aj). Then −cj(bj)− Dj,t ≤0, Dj,t+cj(aj)≤0. Therefore,x∗_j =(c)−1

j (−Dj,t),λ=t,uj=vj=0.

Therefore,x∗,α,λ,u,vsatisfy the KKT conditions (2.24), (2.25), (2.26), (2.27), (2.28), (2.29), (2.30), and (2.31), sox∗∈Pc(D,α,a,b) according to the definition ofPc(D,α,a,b).

The two-way inclusion implies (2.32).

Define the functionsx:Rm_→Rn_,α:Rm_→Rm_by

x(t)=(c)−1_−DT_tc(b)

c_(a), (2.40)

α(t)=D(c)−1−DT_tc(b)

c(a). (2.41)

Then the following corollary holds.

Corollary2.3. Vectorsx∗∈Rn_,α∗_∈Rm_satisfyx∗_=P

c(D,α∗,a,b)if and only if there existst∗∈Rm_{such that}

xt∗=x∗, (2.42)

αt∗=α∗_{. (2.43)}

Proof ofCorollary 2.3follows from the statement of problem (C=_m) and (2.32). It follows fromCorollary 2.3thatx∗=Pc(D,α∗,a,b) can be solved with respect tox∗ for givenα∗by first solving (2.43) fort∗and then calculatingx∗by using (2.42).

LetSbe the set of solutions to (2.43) for a particular value ofα∗:

S=t∈Rm_:α(t)=α∗_{. (2.44)}

According to (2.41), each component ofα(t) is a linear combination of the same set of terms. Each term (c)−_j1[−DT_jt]c

j(bj)

c_j(aj)is a smooth function oftexcept on the pair of break hyperplanes

Aj=t∈Rm:Dj,t= −cj

aj,

Bj=

t∈Rm_:D j,t

= −c

j

bj

. (2.45)

These break hyperplanes are generalizations of breakpoints considered inSection 3.1.

2.3. Problem(C≥). Consider now the problem (C≥) with linear inequality “≥” con-straint (1.2):

min

c(x)=

j∈J

cj

xj

(2.46)

subject to

j∈J

djxj≥α, dj>0, j∈J, (2.47)

Suppose that the following assumptions are satisfied. (1b)aj≤bjfor allj∈J.

(2b)α≤j∈Jdjbj. Otherwise, the constraints (2.47) and (2.48) are inconsistent and

X≥= ∅, whereX≥is defined by (2.47) and (2.48). In addition to this assumption, we suppose that_j∈Jdjaj≤α(see comments after the proof ofTheorem 2.4,Section 2.3).

Leth≥_j, j∈J, be the value ofxj for whichcj(xj)=0. If such a value does not exist, sincec_j(xj) is a monotone nondecreasing function (cj(xj) is convex), we adopth≥j =+∞ for problem (C≥).

Rewrite problem (C≥) in the form

min

c(x)=

j∈J

cjxj

(2.49)

subject to

−

j∈J

djxj≤ −α, dj>0, j∈J, (2.50)

aj≤xj≤bj, j∈J. (2.51)

Since the linear functiond(x)def= −j∈Jdjxj+αis both convex and concave, then (C≥) is a convex optimization problem.

Letλ,λ≥be the Lagrange multipliers associated with (2.2) (problem (C=)) and with (2.47) (problem (C≥)), and letx∗j,x≥j, j∈J, be components of the optimal solutions to (C=), (C≥), respectively. For the sake of simplicity, we useuj,vj,j∈J, instead ofu≥j,v≥j,

j∈J, for the Lagrange multipliers associated withaj≤xj,xj≤bj, j∈J, from (2.51), respectively.

The Lagrangian for problem (C≥) is

Lx,u,v,λ≥=

j∈J

cj

xj

+λ≥ −

j∈J

djxj+α

+

j∈J

uj

aj−xj

+

j∈J

vj

xj−bj

, (2.52)

and the KKT conditions for (C≥) are

cj

x≥j

−λ≥dj−uj+vj=0, j∈J, (2.53)

uj

aj−x≥j

=0, j∈J, (2.54)

vj

x≥_j −bj

=0, j∈J, (2.55)

λ≥ α−

j∈J

djx≥j

=0, λ≥∈R1

+; (2.56)

j∈J

djx≥j ≥α, (2.57)

aj≤x≥j ≤bj, j∈J, (2.58)

We can replace (2.53) and (2.56) by

c_jx≥_j+λ≥dj−uj+vj=0, j∈J, (2.60)

λ≥

j∈J

djx≥j −α

=0, λ≥∈R1

−,dj>0, (2.61)

respectively, where we have redenotedλ≥:= −λ≥_∈R1

−.

Conditions (2.60), (2.54), (2.55), (2.58), and (2.59) withλinstead ofλ≥are among the KKT conditions for problem (C=).

Theorem 2.4 (sufficient condition for optimal solution to problem (C≥)). (i)If λ=

−c

j(x∗j)/dj≤0,j∈Jλ, thenx∗j,j∈J, solve problem(C≥)as well. (ii)Ifλ= −c

j(x∗j)/dj>0,j∈Jλ, thenx≥j,j∈J, defined as follows: (a)x≥_j =bj,j∈Jbλ,

(b)x≥_j =min{bj,h≥j},j∈Jλ,

(c)x≥_j =min{bj,h≥j}for allj∈Jaλsuch thatcj(aj)<0, (d)x≥_j =ajfor allj∈Jaλsuch thatcj(aj)≥0,

solve problem(C≥).

Proof. (i) Letλ= −c

j(x∗j)/dj≤0, j∈Jλ. Sincex∗j, j∈J, satisfy KKT conditions (2.5), (2.6), (2.7), (2.8), (2.9), and (2.10) for problem (C=) as components of optimal solution to (C=), then (2.60), (2.54), (2.55), and (2.57) with equality (and therefore (2.61)), (2.58), and (2.59) are satisfied as well (withλinstead ofλ≥). Since they are the KKT necessary and sufficient conditions for problem (C≥), thenx∗_j,j∈J, solve problem (C≥) as well.

(ii) Letλ= −c

j(x∗j)/dj>0, j∈Jλ. Sincex∗=(x∗j)j∈J is an optimal solution to prob-lem (C=) by the assumption, then KKT conditions (2.5), (2.6), (2.7), (2.8), (2.9), and (2.10) for problem (C=) are satisfied. Ifx≥:=(x≥j)j∈J is an optimal solution to (C≥), thenx≥satisfies (2.60), (2.54), (2.55), (2.61), (2.57), (2.58), and (2.59). Sinceλ >0, then

λcannot play the role ofλ≥in (2.60) and (2.61) becauseλ≥must be anonpositivereal number in (2.60) and (2.61). Thereforex∗j, which satisfy KKT conditions (2.5), (2.6), (2.7), (2.8), (2.9), and (2.10) for problem (C=), cannot play the roles ofx≥j, j∈J, in (2.60), (2.54), (2.55), (2.61), (2.57), (2.58), and (2.59). Hence, in the general case, the equality_j∈Jdjx≥j =αis not satisfied forxj=x≥j. Therefore, in order that (2.61) be sat-isfied,λ≥must be equal to 0. This conclusion helps us to prove the theorem.

Letx≥:=(x≥_j)j∈J be defined as in part (ii) of the statement ofTheorem 2.4. Setλ≥=0;

(1)uj=0,vj= −cj(bj) (≥0 sinceλ >0,dj>0, and according to the definition ofJbλ (2.13)) forj∈Jλ

b;

(2)uj=vj=0 forj∈Jaλsuch thatcj(aj)<0 and forj∈Jλsuch thath≥j < bj; (3)uj=0,vj= −cj(bj) (≥0) forj∈Jλsuch thath≥j ≥bj;

(4)uj=cj(aj)≥0,vj=0 forj∈Jaλsuch thatcj(aj)≥0.

In case (2), we havec_j(aj)<0≡cj(h≥j), thereforeaj≤h≥j =x≥j according to the mono-tonicity ofc_j(xj) and the definition ofx≥j in this case. In case (3), sincebj≤h≥j, then

Consequently, conditions (2.58) and (2.59) are satisfied for alljaccording to (1), (2), (3), and (4).

As we have proved, (2.61) is satisfied withλ≥=0. Since the equality constraint (2.2)

j∈Jdjx∗j =αis satisfied for the optimal solutionx∗to (C=); since the components ofx≥ defined in the statement ofTheorem 2.4(ii), are such that some of them are the same as the corresponding components ofx∗; since some of the components ofx≥, namely those forj∈Jλ

awithcj(aj)<0, are greater than the corresponding componentsx∗j =aj,j∈Jaλ, ofx∗; and sincedj>0, j∈J, then obviously the inequality constraint (2.47) (condition (2.57)) holds forx≥. It is easy to check that other conditions (2.60), (2.54), and (2.55) are also satisfied. Thus,x≥j,j∈J, defined in the statement ofTheorem 2.4(ii), satisfy the KKT conditions for problem (C≥). Therefore,x≥is an optimal solution to problem (C≥).

According toTheorem 2.4, the optimal solution to problem (C≥) is obtained by using the optimal solution and optimal value of the Lagrange multiplierλfor problem (C=). That is why we suppose that_j∈Jdjaj≤αin addition to assumption (2b) (seeStep 1of Algorithm 3.2,Section 3.4), as we assumed this in assumption (2a) for problem (C=).

3. Algorithms

3.1. Analysis of the optimal solution to problem(C=). Since the optimal solutionx∗ to problem (C=) depends onλ, we consider the components ofx∗as functions ofλfor differentλ∈R1_:

xj(λ)=         

aj, j∈Jaλ,

bj, j∈Jbλ,

x∗j :cj

x∗j

+λdj=0, j∈Jλ.

(3.1)

Functionsxj(λ),j∈J, are piecewise linear, monotone, piecewise differentiable functions ofλ, with two breakpoints atλ= −c

j(aj)/djandλ= −cj(bj)/dj. Let

δ(λ)def=

j∈Jλ a

djaj+

j∈Jλ b

djbj+

j∈Jλ

djxj(λ)−α. (3.2)

According touj=vj=0, j∈Jλ, condition (2.5) becomes

cj

xj(λ)

+λdj=0, j∈Jλ. (3.3)

If we differentiate both sides of each of these expressions with respect to λ(using that

c_j(xj),j∈Jλ, exist by assumption;xj(λ),j∈Jλ, exist becausexj(λ) are defined byxj(λ)=

x∗j such thatcj(x∗j) +λdj=0 forj∈Jλ), we obtain

cj

xj(λ)

Therefore,

x_j(λ)= − dj

c_jxj(λ), j∈J

λ_{, (3.5)}

and sincec_j(xj)≥0, j∈J, as the second derivatives of convex differentiable functions;

dj>0 by the assumption, thenxj(λ)<0, j∈Jλ. (If we assume thatcj(xj(λ))=0, then

dj=0 according to (3.4). However,dj>0, j∈J, by the assumption, a contradiction.) Consequently,

δ(λ)≡

j∈Jλ

djxj(λ)<0, (3.6)

whenJλ_{= ∅, andδ(λ)=0 whenJ}λ_{= ∅. Hence,δ(λ) isa monotone nonincreasing} func-tionofλ∈R1_.

Using equationδ(λ)=0, whereδ(λ) is defined by (3.2), we are always able to deter-mineλas an implicit function ofx:λ=λ(x), becauseδ(λ)<0 whenJλ_{= ∅according to} (3.6) (it is important thatδ(λ)=0). Moreover, sinceδ(λ) is a linear function ofx(λ), it is always possible to obtain aclosed-form expressionofλ. It turns out that for our purpose, without loss of generality, we can assume thatδ(λ)=0, that is,δ(λ) depends ofλ, which means thatJλ_{= ∅(see the third paragraph ofRemark 3.3).}

At iterationk of the implementation of algorithms, denote byλ(k) _{the value of the} Lagrange multiplier associated with the constraint (2.2) ((2.47), resp.), byα(k)_the right-hand side of (2.2) ((2.47), resp.); byJ(k)_{, J}λ(k)

a ,J_bλ(k),Jλ(k) the current setsJ,Jaλ,Jbλ,Jλ, respectively.

3.2.Algorithm 3.1(for problem(C=)). According toTheorem 2.1and the preliminary analysis, we can suggest the following algorithm for solving problem (C=) with strictly convex differentiable functionscj(xj), seeAlgorithm 3.1.

Remark 3.1. To avoid a possible “endless loop” in programingAlgorithm 3.1, the criterion ofStep 5to go toStep 8at iterationkusually is notδ(λ(k)_{)=0 butδ(λ}(k)_{)∈[−ε,ε] where}

ε >0 is some tolerance value up to which the equalityδ(λ∗)=0 (i.e.,_j∈Jdjx∗j =α) must be satisfied.

3.3. Convergence and complexity ofAlgorithm 3.1. Theorem 3.2states convergence of Algorithm 3.1, that is, “convergence” ofλ(k)_,Jλ(k)_,Jλ(k)

a ,J_bλ(k), generated byAlgorithm 3.1, to the optimalλ,Jλ_,Jλ

a,JbλfromTheorem 2.1, respectively.

Theorem3.2. Letλ(k)_{be the sequence generated byAlgorithm 3.1. Then,}

(i)ifδ(λ(k)_{)>0, thenλ}(k)_≤λ(k+1)_; (ii)ifδ(λ(k)_{)<0, thenλ}(k)_≥λ(k+1)_.

Proof. Denote byx(_jk)the components ofx(k)_=(x

j)j∈J(k)at iterationkof implementation

Step 1(initialization). J:= {1,...,n},k:=0,J(0)_:=J,α(0)_:=α,n(0)_:=n,Jλ a:= ∅,

J_bλ:= ∅, initializeh=_j,j∈J. If_j∈Jdjaj≤α≤

j∈Jdjbj, go toStep 2, else go to Step 9.

Step 2. Jλ(k)_:=J(k)_{. Calculateλ}(k)_{by using the closed-form expression ofλ,}

determined from the equality constraint_j∈Jλ(k)d_jx_j=α(k), wherex_jare given by

(2.14). Go toStep 3.

Step 3. Construct the setsJaλ(k),J_bλ(k),Jλ(k)through (2.12), (2.13), and (2.14) (with

j∈J(k)_{instead ofj∈J) and find their cardinalities|J}λ(k)

a |,|J_bλ(k)|,|Jλ(k)|, respectively. Go toStep 4.

Step 4. Calculate

δλ(k):=

j∈Jλa(k)

djaj+

j∈Jbλ(k)

djbj+

j∈Jλ(k)

djx∗j −α(k), (3.7)

wherex∗j,j∈Jλ(k), are determined from (2.14) withλ=λ(k). Go toStep 5. Step 5. Ifδ(λ(k)_{)=0 orJ}λ(k)_{= ∅, thenλ:=λ}(k)_,Jλ

a:=Jaλ∪Jaλ(k),J_bλ:=J_bλ∪J_bλ(k),

Jλ_:=Jλ(k)_{, go toStep 8;}

else ifδ(λ(k)_{)>0, go toStep 6;} else ifδ(λ(k)_{)<0, go toStep 7.}

Step 6. x∗_j :=ajforj∈Jaλ(k),α(k+1):=α(k)−_j∈Jλ(k)

a djaj,J

(k+1)_:=J(k)_\Jλ(k) a ,

n(k+1)_:=n(k)_{− |J}λ(k)

a |,J_aλ:=J_aλ∪Jaλ(k),k:=k+ 1. Go toStep 2. Step 7. x∗_j :=bjforj∈Jbλ(k),α(k+1):=α(k)−

j∈Jbλ(k)djbj,J

(k+1)_:=J(k)_\Jλ(k) b ,

n(k+1)_:=n(k)_{− |J}λ(k)

b |,Jbλ:=Jbλ∪Jλ (k)

b ,k:=k+ 1. Go toStep 2.

Step 8. x∗j :=ajforj∈Jaλ;x∗j :=bjforj∈Jbλ; assignx∗j the value determined from (2.14) forj∈Jλ_{. Go toStep 10.}

Step 9. Problem (C=) has no optimal solution becauseX== ∅. Step 10. End.

Algorithm3.1

(i) Letδ(λ(k)_{)>0. UsingStep 6ofAlgorithm 3.1, which is performed whenδ(λ}(k)_)>0, we get

j∈Jλ(k+1)

djx(jk)≡

j∈J(k+1)

djx(jk)=

j∈J(k)_\Jλ(k)

a

djx(jk)=α(k)−

j∈Jaλ(k)

djx(jk). (3.8)

Let j∈Jaλ(k). According to the definition (2.12) ofJaλ(k)and relationλ(k)dj= −cj(x (k) j ), we have

−c

j

aj

dj ≤λ

(k)_{= −}c

j

x(_jk)

Multiplying this inequality by−dj<0, we obtaincj(aj)≥cj(x (k)

j ). Hence,aj≥x(jk), j∈

Jaλ(k), in accordance with monotonicity ofc_j(xj).

Taking into consideration (3.8),dj>0,aj≥x(jk),j∈Jλ (k)

a , andStep 6, we get

j∈Jλ(k+1)

djx(jk)=α(k)−

j∈Jaλ(k)

djx(jk)≥α(k)−

j∈Jaλ(k)

djaj=α(k+1)=

j∈Jλ(k+1)

djx(jk+1). (3.10)

Sincedj>0, j∈J, then there exists at least onej0∈Jλ(k+1)such thatx(_jk0)≥x (k+1) j0 . Then

λ(k)_{= −}c

j0

x(_jk0)

dj0

≤ −c

j0

x(_jk0+1)

dj0

=λ(k+1)_{, (3.11)}

where we have used the relationship (2.14) betweenλ(k)_andx(k)

j for j∈Jλ(k)according toStep 2ofAlgorithm 3.1, the fact that−c

j0(xj) is a monotone nonincreasing function,

anddj>0, j∈J.

The proof of part (ii) is omitted because it is similar to that of part (i).

Remark 3.3. Since we do not know the optimal value ofλwhich is involved in the state-ment ofTheorem 2.1, we approximate the value ofλuntil we obtain the optimal value of

λat the last iteration of algorithm performance. In order to determine the current value

λ(k)_{ofλat each iteration (including the initial value), we assume thatJ}λ(k)_=J(k)_{at the} beginning of the corresponding iteration (Step 2).

Theorem 3.2, definitions ofJλ

a (2.12),Jbλ (2.13), and Steps6and7 ofAlgorithm 3.1 allow us to assert that the values ofλ(k)_{,k=0, 1,..., calculated atStep 2, are such that}

j∈Jaλ(k)implies thatj∈Jaλ(k+1),j∈Jbλ(k)implies thatj∈Jλ (k+1)

b , and sinceJλ(k)is reduced (Steps6and7), thenj∈Jλ(k+1)_{implies thatj∈J}λ(k)_{; that is, we haveJ}λ(k)

a ⊆Jaλ(k+1),J_bλ(k)⊆

J_bλ(k+1), andJλ(k)_⊇Jλ(k+1)_{. This means that if j belongs to current index setJ}λ(k) a , then j belongs to next index setJaλ(k+1)and, therefore, to the “optimal” index setJaλ according toTheorem 3.2 and definition (2.12); the same holds true about the setsJ_bλ(k) and J_bλ

(2.13). Therefore,λ(k)_{converges to the optimal valueλfromTheorem 2.1andJ}λ(k) a ,J_bλ(k),

Jλ(k) _{“converge” toJ}λ

a,Jbλ,Jλ, respectively. This means that calculation ofλ, operations

x∗_j :=aj, j∈Jaλ(k)(Step 6),x∗j :=bj, j∈Jbλ(k)(Step 7), and the construction ofJaλ,Jbλ,Jλ are in accordance withTheorem 2.1. The final setsJλ

a,Jbλ,Jλare constructed atStep 1or atStep 5(whenδ(λ(k)_{)=0 orJ}λ(k)_{= ∅) of iterationk, wherekis the number of the last} iteration of algorithm performance.

Since at the beginning ofAlgorithm 3.1, we haveJλ(0)_{:=J(Steps1,2) and sinceJ}λ(k)_⊇

Jλ(k+1)_{, thenJ}λ(k)_{= ∅for allk≤k}

0, wherek0is some nonnegative integer. If we obtain

Jλ(k0)= ∅_{, this would mean thatJ}λ(k0)

a ∪J_bλ(k0)=J, that is, the problem has been already solved at iterationk0, andδ(λ(k0))=const.

At each iteration,Algorithm 3.1calculates the value of at least one variable (Steps6,7, and8) and at each iteration, we solve a problem of the form (C=) butof less dimension (Steps2,3,4,5,6, and7). Therefore,Algorithm 3.1is finite and it converges with at most

n= |J|iterations, that is, the iteration complexity ofAlgorithm 3.1isᏻ(n).

Step 1 takes time ᏻ(n). The calculation of x(_jk), j∈J, and λ(k) _{requires ᏻ(n) time} (Step 2). Step 3takesᏻ(n) time because of the construction of Jaλ(k),Jbλ(k),Jλ(k).Step 4 also requiresᏻ(n) time andStep 5requires constant time. Each of Steps6,7, and8takes time which is bounded byᏻ(n) because at these steps, we assign some of thexj’s optimal value, and since the number of allxj’s isn, then Steps6,7, and8take timeᏻ(n). Hence, Algorithm 3.1hasᏻ(n2_{) running time and it belongs to the class of strongly polynomially} bounded algorithms.

As the computational experiments show (Section 5), the number of iterations of the algorithm performance is not only at mostnbut it is much, much less thannfor large

n. In fact, this number does not depend onnbut only on the three index sets defined by (2.12), (2.13), and (2.14). In practice,Algorithm 3.1hasᏻ(n) running time.

Consider thefeasibilityofx∗=(x∗_j)j∈Jgenerated byAlgorithm 3.1.

Components x∗_j =aj, j∈Jaλ, and x∗j =bj, j ∈Jbλ, obviously satisfy (2.3). Using

−c

j(bj)/dj< λ≡ −cj(x∗j)/dj<−cj(aj)/dj,j∈Jλ, anddj>0,j∈J, it follows thatcj(aj)<

c_j(x∗_j)< c_j(bj), j∈Jλ. Therefore,aj≤x∗j ≤bjforj∈Jλas well according to the mono-tonicity ofc_j(xj). Hence, allx∗j,j∈J, satisfy (2.3).

In the sequel, since at each iterationλ(k)_{is determined from the “current” equality} constraint (2.2) (Step 2ofAlgorithm 3.1) and sincexj, j∈J, are determined in accor-dance withλ(k)_{at each iteration (Steps5,6,7, and8ofAlgorithm 3.1), thenx}∗_satisfies

(2.2) as well.

Thereforex∗, obtained byAlgorithm 3.1, is feasible for (C=), which is an assumption ofTheorem 2.1.

3.4.Algorithm 3.2(for problem(C≥)). Algorithm 3.2 for solving problem (C≥) with strictly convex differentiable functionscj(xj) is based onTheorem 2.4andAlgorithm 3.1 (seeAlgorithm 3.2).

SinceAlgorithm 3.2is based onTheorem 2.4andAlgorithm 3.1and since the “itera-tive” Steps2,3,4,5,6, and7of Algorithms3.1and3.2are the same, then “convergence” ofAlgorithm 3.2follows fromTheorem 3.2as well. Because of the same reason, compu-tational complexity ofAlgorithm 3.2is the same as that ofAlgorithm 3.1.

4. Extensions

4.1. Theoretical aspects. Up to now, we requireddj>0, j∈J, in (2.2) and (2.47) of problems (C=) and (C≥), respectively. However, if it is allowed thatdj=0 for somej∈J in problems (C=) and (C≥), then for such indicesj, we cannot construct the expressions

−c

Step 1(initialization). J:= {1,...,n};k:=0;J(0)_:=J;α(0)_:=α,n(0)_:=n;Jλ a:= ∅,

J_bλ:= ∅, initializeh≥_j,j∈J. If_j∈Jdjaj≤α≤

j∈Jdjbj, go toStep 2, else go to Step 9.

Steps2,3,4,5,6, and7are the same as Steps2,3,4,5,6, and7ofAlgorithm 3.1, respectively.

Step 8. Ifλ≤0, thenx≥_j :=ajforj∈Jaλ,x≥j :=bjforj∈Jbλ, assignx≥j the value determined through (2.14) forj∈Jλ_{, go toStep 10;}

else ifλ >0, then

x≥_j :=bjforj∈Jbλ,

x≥_j :=min{bj,h≥j}forj∈Jλ, ifj∈Jλ

aandcj(aj)<0, thenx≥j :=min{bj,h≥j}; else ifj∈Jλ

a andcj(aj)≥0, thenx≥j :=aj; go toStep 10.

Step 9. Problem (C≥) has no optimal solution becauseX≥= ∅or there do not exist

x∗_j ∈[aj,bj], j∈J, such that

j∈Jdjx∗j =α. Step 10. End.

Algorithm3.2

Denote

Z0=j∈J:dj=0. (4.1)

Here, “0” denotes the “computer zero.” In particular, whenJ=Z0 andα=0, thenX=(or

X≥) is defined only by (2.3) (by (2.48), resp.).

Theorem 4.1 (characterization of the optimal solution to problem (C=): an extended

version). Problem(C=)can be decomposed into two subproblems:(C1=)for j∈Z0and (C2=)for j∈J\Z0. The optimal solution to(C1=)is

x∗_j =

        

aj, j∈Z0,h=j ≤aj,

bj, j∈Z0,h=j ≥bj,

h=_j, j∈Z0,aj< h=j < bj.

(4.2)

The optimal solution to(C2=)is given by (2.12), (2.13), and (2.14) withJ:=J\Z0.

Proof. Necessity. Letx∗=(x∗j)j∈Jbe an optimal solution to (C=). (1) Let j∈Z0, that is,dj=0 for thisj. The KKT conditions are

cj

x∗j

−uj+vj=0, j∈Z0, (4.3)

and (2.6), (2.7), (2.8), (2.9), and (2.10).

(b) Ifx∗_j =bj, thenuj=0,vj≥0 according to (2.6), (2.7), and (2.8). Therefore, (4.3) implies thatcj(x∗j)= −vj≤0≡cj(h=j). Using monotonicity ofcj(xj), we obtainx∗j =

bj≤h=j.

(c) Ifaj< x∗j < bj, thenuj=vj=0 according to (2.6) and (2.7). Therefore, (4.3) im-plies that−c_j(x∗_j)=0, that is,x∗_j =h=_j according to definition ofh=_j.

(2) Components of the optimal solution to (C2=) are obtained by using the approach as that of the necessity part of the proof ofTheorem 2.1but with the reduced index set

J:=J\Z0.

Sufficiency. Conversely, letx∗∈X=and components ofx∗satisfy (4.2) forj∈Z0, and (2.12), (2.13), and (2.14) forj∈J\Z0. Set

λ=0; uj=vj=0 foraj< x∗j < bj, j∈Z0;

uj=cj

aj (≥0), vj=0 forx∗j =aj, j∈Z0;

uj=0, vj= −cj

bj

(≥0) forx∗_j =bj, j∈Z0;

λ= −c

j

x∗j

dj ; uj=vj=0 foraj< x

∗

j < bj, j∈J\Z0;

uj=cj

aj

+λdj (≥0), vj=0 forx∗j =aj, j∈J\Z0;

uj=0, vj= −cj

bj

−λdj (≥0) forx∗j =bj, j∈J\Z0.

(4.4)

It can be verified thatx∗,λ,uj,vj, j∈J, satisfy the KKT conditions (2.5), (2.6), (2.7), (2.8), (2.9), and (2.10). Thenx∗with components (4.2) for j∈Z0, and (2.12), (2.13), and (2.14) withJ:=J\Z0 is an optimal solution to problem (C=)=(C1=)∪(C2=).

Thus, with the use ofTheorem 4.1, we can express componentsx∗_j, j∈Z0, of the optimal solution to (C=) (and therefore those to problem (C≥))withoutthe necessity of constructing the expressions−c

j(aj)/djand−cj(bj)/djwithdj=0. SinceTheorem 2.4andAlgorithm 3.2are based on the sets of indicesJλ

a,Jbλ,Jλof prob-lem (C=), thenTheorem 4.1solves the problem of decomposition of problem (C≥) as well.

With the use of setZ0, we can deduce the following about checking whether the feasi-ble region is empty or nonempty whenJ=Z0 for problems (C=) and (C≥).

When J=Z0, aj≤bj, j∈J,α=0, the corresponding feasible regions are always nonempty and it is not necessary to check anything else in this case.

4.2. Computational aspects. Algorithms3.1and3.2are also applicable in cases when

aj= −∞for some j∈Jand/orbj= ∞for some j∈J. However, if we use the computer values of−∞and +∞atStep 1of the algorithms to check whether the corresponding feasible region is empty or nonempty and atStep 3in the expressions−c_j(xj)/dj with

inconvenience and dependence on the data of the particular problems. To avoid these difficulties and to take into account the above discussion, it is convenient to do the fol-lowing.

Construct the sets of indices

SVN=j∈J\Z0 :aj>−∞,bj<+∞

,

SV1=j∈J\Z0 :aj>−∞,bj=+∞

, SV 2=j∈J\Z0 :aj= −∞,bj<+∞

,

SV=j∈J\Z0 :aj= −∞,bj=+∞

.

(4.5)

It is obvious thatZ0∪SV∪SV 1∪SV 2∪SVN=J, that is, the setJ\Z0 is partitioned into the four sets SVN, SV 1, SV 2, SV, defined above.

When programming the algorithms, we use computer values of−∞and +∞for con-structing the sets SVN, SV 1, SV 2, SV.

In order to construct the setsJλ

a,Jbλ,Jλ without the necessity of calculating the values

−c

j(xj)/djwithxj= −∞orxj=+∞, except for the setsJ,Z0, SV, SV 1, SV 2, SVN, we need some subsidiary sets defined as follows.

For SVN,

JλSVN=

j∈SVN :−c

j

bj

dj < λ <−

c_jaj

dj

,

JaλSVN=

j∈SVN :λ≥ −c j aj dj ,

JλSVN

b =

j∈SVN :λ≤ −c j bj dj . (4.6)

For SV 1,

JλSV 1=

j∈SV 1 :λ <−c j aj dj ,

JaλSV 1=

j∈SV 1 :λ≥ −c j aj dj . (4.7)

For SV 2,

JλSV 2₌

j∈SV 2 :λ >−c j bj dj ,

J_bλSV 2=

j∈SV 2 :λ≤ −c j bj dj . (4.8) For SV,

Then,

Jλ:=JλSVN∪JλSV 1∪JλSV 2∪JλSV,

Jaλ:=JaλSVN∪JaλSV 1,

J_bλ:=J_bλSVN∪J_bλSV 2.

(4.10)

We use the setsJλ_,Jλ

a,Jbλ in (4.10) as the corresponding sets with the same names in Algorithms3.1and3.2.

The reason to construct namely the sets (4.5), (4.6), (4.7), (4.8), and (4.9) is the fol-lowing.

Ifj∈SVN, then none of theaj’s is equal to−∞and none of thebj’s is equal to +∞. That is why there is not any peculiarity of the described type for such indicesj.

Ifj∈SV 1, that is, ifaj>−∞andbj=+∞, thenvj=0, j∈SV 1, according to (2.7) for problem (C=) and according to (2.55) for problem (C≥). Then j∈Jλ

a orj∈Jλusing the same reasoning as in (a) and (c) from the proof (necessity part) ofTheorem 2.1. Therefore, it is sufficient to consider only sets of the type ofJλ

a andJλfor SV 1, and we have denoted these sets byJλSV 1

a andJλSV 1in (4.7), respectively.

Similarly, if j∈SV 2, thenbj<+∞andaj= −∞for these j’s. Thenuj=0 according to (2.6) for problem (C=) or according to (2.54) for problem (C≥). Hencej∈Jλ

borj∈Jλ according to cases (b) and (c) of the proof (necessity part) ofTheorem 2.1. That is why it is sufficient to consider only sets of the typeJ_bλandJλ_{for SV 2. We have denoted these} sets byJλSV 2

b andJλSV 2in (4.8), respectively.

If j∈SV, thenaj= −∞andbj=+∞. Therefore,uj=vj=0 according to (2.6) and (2.7) for problem (C=), and according to (2.54) and (2.55) for problem (C≥). Hence

j∈Jλ _{according to (c) from the proof (necessity part) ofTheorem 2.1. Therefore j∈} SV implies that j∈Jλ_{, and we have denoted byJ}λSV _{the set{j∈SV :−c}

j(bj)/dj< λ <

−c

j(aj)/dj}in this case.

SinceTheorem 2.4(sufficient condition for solution to (C≥)) is based on the index sets for problem (C=) fromTheorem 2.1, these conclusions are also valid for problem (C≥).

The assumption thatdj≥0, j∈J, for problems (C=) and (C≥) helps us to draw the following conclusions.

About problem(C=). (i) If SVN=J\Z0, that is, ifaj’s and bj’s are finite for all j∈J which are involved in (2.2) withdj=0, then the checking whetherXis nonempty is

j∈SVN

djaj≤α≤

j∈SVN

djbj. (4.11)

(ii) Else if SV 1∪SVN=J\Z0, that is, if allaj’s are finite but some of (or all)bj’s are equal to +∞for the variables which are involved in (2.2), then the checking is whether

j∈J\Z0

djaj≤α (4.12)

(11_{) (initialization)J:= {1,...,n};k:=0;J}(0)_:=J;α(0)_:=α,n(0)_:=n;Jλ a:= ∅,

J_bλ:= ∅, initializeh=_j,j∈J. Construct the setZ0. Ifj∈Z0, then

ifh=_j ≤aj, thenx∗j :=aj; else ifh=j ≥bj, thenx∗j :=bj; else ifaj< h=j < bj, thenx∗j :=h=j.

IfJ=Z0 andα=0, go toStep 10, else ifJ=Z0 andα=0, go toStep 9. SetJ:=J\Z0,J(0)_:=J,n(0)_{:=n− |Z0|,α}(0)_:=α−

j∈Z0djx∗j ≡α. Construct the sets SVN, SV 1, SV 2, SV.

If SVN=J, then if_j∈Jdjaj≤α≤

j∈Jdjbj, go toStep 2; else go toStep 9(feasible regionX=is empty) else if SV 1∪SVN=J, then

if_j∈Jdjaj≤α, go toStep 2;

else go toStep 9(feasible regionX=is empty) else if SV 2∪SVN=J, then

ifα≤_j∈Jdjbj, go toStep 2;

else go toStep 9(feasible regionX=is empty);

else if SV= ∅, go toStep 2(feasible regionX=is always nonempty). (31_{) Construct the setsJ}λSVN_,JλSVN

a ,JbλSVN,JλSV 1,JaλSV 1,JλSV 2,JbλSV 2,JλSV(withJ(k) instead ofJ).

Construct the setsJaλ(k),J_bλ(k),Jλ(k)by using (4.10) and find their cardinalities. Go to Step 4.

Algorithm4.1. AboutAlgorithm 3.1.

(iii) Else if SV 2∪SVN=J\Z0, that is, if allbj’s are finite but some of (or all)aj’s are equal to−∞for variables which are involved in (2.2), then the checking is whether

α≤

j∈J\Z0

djbj (4.13)

and it is not necessary to check whether_j∈J\Z0djaj≤αin this case.

(iv) Else if SV= ∅, that is, when there exists at least one variablexjwhich is involved in (2.2) withaj= −∞andbj=+∞, thenX== ∅and it is not necessary to check anything else in this case.

Similarly we can treat problem (C≥).

With the use of results of this section, Steps1and3ofAlgorithm 3.1can be modified as follows (seeAlgorithm 4.1), respectively.

Similarly, we can modify Steps1and3ofAlgorithm 3.2.