R E S E A R C H
Open Access
Solving nonlinear programming problems
with unbounded non-convex constraint sets
via a globally convergent algorithm
Menglong Su
1*, Jianhua Li
1and Zhonghai Xu
2*Correspondence:
1School of Mathematical Sciences,
Luoyang Normal University, Luoyang, 471022, P.R. China Full list of author information is available at the end of the article
Abstract
In this paper, we propose a set of unbounded conditions, under which we are able to solve nonlinear programming problems in a class of unbounded non-convex sets via the combined homotopy interior point algorithm. We also obtain the global
convergence of the combined homotopy interior point algorithm and analyze the implementation of this algorithm in detail.
Keywords: unbounded conditions; nonlinear programming problems; non-convex sets
1 Introduction
Consider the constrained minimization problem:
min x∈Rnf(x)
s.t.ui(x)≤, i= , . . . ,m, ()
vj(x) = , j= , . . . ,l,
wheref :Rn→R,ui:Rn→R,i= , . . . ,mandvj:Rn→R,j= , . . . ,lare three times
continuously differentiable.
We call a pointx∗is a Karush-Kuhn-Tucker (K-K-T) point of () andy∗,z∗are the cor-responding Lagrangian multiplier vectors if (x∗,y∗,z∗) satisfies
∇f(x) +∇u(x)y+∇v(x)z= ,
v(x) = ,
Yu(x) = , u(x)≤, y≥,
()
where y∈Rm, z ∈Rl, ∇f(x) = (∂f(x)/∂x)T ∈Rn, ∇u(x) = (∇u
(x), . . . ,∇um(x))∈Rn×m,
∇v(x) = (∇v(x), . . . ,∇vl(x)) ∈ Rn×l, u(x) = (u(x), . . . ,um(x))T ∈ Rm, v(x) = (v(x), . . . ,
vl(x))T∈RlandY=diag(y)∈Rm×m.
Since Kellogget al.(see []) and Smale (see []) proposed the notable homotopy method, this method has become a powerful tool in dealing with various nonlinear problems, for
example, zeros or fixed points of maps (see [–],etc.and the references therein). How-ever, the homotopy method has seldom been touched upon in constrained optimization until , Megiddo (see []) and Kojimaet al.(see []) discovered the Karmarkar interior point method for linear programming was a kind of path-following method. From then on, the central path-following methods for mathematical programming have become an active research subject. Furthermore, it was extended to convex nonlinear programming problems recently (see [–],etc.). But all their convergence results were obtained under the assumptions that the logarithmic barrier function is strictly convex and the solution set is nonempty and bounded.
Recently, a combined homotopy interior point method (denoted as CHIP method for convenience) for nonlinear programming problems was presented in [, ] (detailed ab-stract of them was announced in []). In [], compared with the central path-following methods, the authors removed the convexity of the logarithmic barrier function and the boundedness and nonemptiness of the solution set. This shows that the CHIP method can solve the problems that interior path-following methods cannot solve. In [], by tak-ing a piecewise technique, under the commonly used conditions, the polynomiality of the CHIP method was given, which shows that the efficiency of the CHIP method is also very well. In [], we introducedCfunctionsα
i(x)∈Rn,i= , . . . ,mandβj(x)∈Rn,j= , . . . ,l
to extend the results in [, ] to more general non-convex sets. However, there are no re-sults reported in the literature about the work in [] extended to unbounded non-convex sets. In this paper, we attempt to complete this work. To this end, by using some inequal-ity techniques and the ideas of infinite solutions which were introduced in [, ], we develop a set of new unboundedness conditions. Under these conditions, we obtain the global convergence results of the CHIP method and therefore extend the work in [] to unbounded non-convex sets.
2 Main results
In this section, the nonnegative and positive orthants ofRm are denoted asRm
+ andRm++, respectively. We also denote byB(x) ={i∈ {, . . . ,m}:ui(x) = }the active set atx. In
ad-dition, letX={x∈Rn:u
i(x)≤,i= , . . . ,m,vj(x) = ,j= , . . . ,l}be the feasible set, let X={x∈Rn:ui(x) < ,i= , . . . ,m,vj(x) = ,j= , . . . ,l}be the strictly feasible set and let
∂X=X\Xbe the boundary set ofX.
In [], to solve (), the following homotopy was constructed:
Hw,w(),λ
= ⎛ ⎜ ⎝
( –λ)[∇f(x) +∇u(x)y+λξ(x)y] + [∇v(x) +λ(η(x) –∇v(x))]z+λ(x–x())
v(x)
Yu(x) –λY()u(x())
⎞ ⎟ ⎠
= , ()
where w= (x,y,z)∈Rn+m+l,w() = (x(),y(), )∈X×Rm
++ ×Rl, and Y() =diag(y())∈
Rm×m.
k→ ∞, and for any givenζ ∈X, there existy(k)∈Rm
+ andz(k)∈Rlsuch that
lim k→∞
ζ–x(k)T∇fx(k)+∇ux(k)y(k)+∇vx(k)z(k)≥.
Then we assume that there exist smooth mappingsξi(x)∈R,i= , . . . ,mandηj(x)∈R, j= , . . . ,lsuch that:
(A) Xis nonempty; nonlinear programming problems have no infinite solutions.
(A) ∀x∈X, if
i∈B(x)
yi∇gi(x) +αiξi(x)
+∇v(x)z+η(x)β= , yi≥,αi≥,z∈Rl,β∈Rl,
thenyi= ,αi= andz=β= ,∀i∈B(x).
(A) ∀x∈X, we have
x+
i∈B(x)
yiξi(x) +zη(x) :yi≥,i∈B(x),z∈Rl
∩X={x},
whereη(x) = (η(x), . . . ,ηl(x)).
(A) ∀x∈X,∇v(x)is of full of column rank and∇v(x)Tη(x)is nonsingular.
However, only applying the infinite solution technique to the items∇f(x),∇u(x), and ∇v(x), we cannot extend the results in [] to unbounded cases because of the existence of the itemsξ(x) andη(x). So for the itemsξ(x) andη(x), we need to use other techniques. In this paper, we still need the following assumption:
(A) ∀ζ∈X,
ui(ζ)T–ui(x)T≥(ζ–x)Tξi(x)yi, i= , . . . ,m, v(ζ)T–v(x)T≥(ζ–x)Tη(x).
For a givenw(), rewriteH(w,w(),λ) asH
w()(w,λ). The zero-point set ofHw()is
Hw–()() =(w,λ)∈X×R+m×Rl×(, ] :Hw()(w,λ) = .
The inverse image theorem tells us that, if is a regular value of the mapHw(), then
H–
w()() consists of some smooth curves. The regularity ofHw()can be obtained by the following lemma.
Lemma .(Parameterized Sard theorem) Let V⊂Rn,U⊂Rmbe open sets,and :V×
U→R(k) a Cr map,where r>max{,m–k}.If∈R(k)is a regular value of ,then for
almost all a∈V, is a regular value of a≡ (a,·).
Lemma . Let H be defined as in().In addition, let assumptions(A)-(A)hold,let
ui(x),i= , . . . ,m and vj(x),j= , . . . ,l be Cfunctions,and letξi(x),i= , . . . ,m andηj(x), j= , . . . ,l be Cfunctions.Then,for almost all w()∈X×Rm
++×Rl, is a regular value
Lemma . Let H be defined as in().In addition, let assumptions(A)-(A)hold,let
ui(x),i= , . . . ,m and vj(x),j= , . . . ,l be Cfunctions,and letξi(x),i= , . . . ,m andηj(x), j= , . . . ,l be Cfunctions.Then,for almost all w()∈X×Rm
++×Rl,the projection of the
smooth curvew()onto the x-plane is bounded.
Proof If not, then there exists a sequence of points{(x(k),y(k),z(k),λ
k)}∞k=such that x(k) → ∞ask→ ∞.
It is easy to show that the following inequality holds:
x–ζ –x()–ζ≤(x–ζ)Tx–x(). ()
By the homotopy equation (), we have
Hw(k),w(),λ
k
= ⎛ ⎜ ⎜ ⎜ ⎝
( –λk)[∇f(x(k)) +∇u(x(k))y(k)+λkξ(x(k))(y(k))] +λk(x(k)–x())
+ [∇v(x(k)) +λ
k(η(x(k)) –∇v(x(k)))]z(k) v(x(k))
Y(k)u(x(k)) –λ
kY()u(x())
⎞ ⎟ ⎟ ⎟ ⎠
= . ()
Multiplying the first equation in () by (x(k)–ζ)T, we get
( –λk)
x(k)–ζT∇fx(k)+ ( –λk)
x(k)–ζT∇ux(k)y(k)
+ ( –λk)λk
x(k)–ζTξx(k)y(k)+x(k)–ζT∇vx(k)z(k)
+x(k)–ζTλk
ηx(k)–∇vx(k)z(k)+λk
x(k)–ζTx(k)–x()= , ()
i.e.,
λk
x(k)–ζTx(k)–x()
= –( –λk)
x(k)–ζT∇fx(k)– ( –λk)
x(k)–ζT∇ux(k)y(k)
– ( –λk)λk
x(k)–ζTξx(k)y(k)
–x(k)–ζT∇vx(k)+λk
ηx(k)–∇vx(k)z(k)
= –( –λk)
x(k)–ζT∇fx(k)– ( –λk)
x(k)–ζT∇ux(k)y(k)
– ( –λk)λk
x(k)–ζTξx(k)y(k)– ( –λk)
x(k)–ζT∇vx(k)z(k)
–λk
x(k)–ζTηx(k)z(k). ()
So
λkx(k)–ζ
–x()–ζ
≤λk
x(k)–ζTx(k)–x()
= –( –λk)
x(k)–ζT∇fx(k)– ( –λk)
– ( –λk)λk
x(k)–ζTξx(k)y(k)– ( –λk)
x(k)–ζT∇vx(k)z(k)
– λk
x(k)–ζTηx(k)z(k)
= –( –λk)
x(k)–ζT∇fx(k)+∇ux(k)y(k)+∇vx(k)z(k)
+ ( –λk)λk
ζ–x(k)Tξx(k)y(k)+ λk
ζ–x(k)Tηx(k)z(k)
≤–( –λk)
x(k)–ζT∇fx(k)+∇ux(k)y(k)+∇vx(k)z(k)
+ ( –λk)λk
u(ζ)Ty(k)–ux(k)Ty(k)+ λk
v(ζ)Tz(k)–vx(k)Tz(k). ()
Sinceu(ζ)T≤ andy(k)≥, thenu(ζ)Ty(k)≤. Besides, by the third equation in (), we haveu(x(k))Ty(k)=λ
ku(x())Ty(), thus the following inequality holds:
λkx(k)–ζ
–x()–ζ
≤–( –λk)
x(k)–ζT
∇fx(k)+∇ux(k)y(k)+∇vx(k)z(k)
– ( –λk)λku
x(k)Ty(k)
= –( –λk)
x(k)–ζT∇fx(k)+∇ux(k)y(k)+∇vx(k)z(k)
– ( –λk)λku
x()Ty(). ()
Then by (), we have
ζ–x(k)T∇fx(k)+∇ux(k)y(k)+∇vx(k)z(k)
≥ λk
( –λk)
x(k)–ζ–x()–ζ+λkux()Ty(). ()
When x(k) → ∞, by (), we have
lim k→∞
ζ–x(k)T∇fx(k)+∇ux(k)y(k)+∇vx(k)z(k)
≥ lim k→∞
λk
( –λk)
x(k)–ζ–x()–ζ+λ
ku
x()Ty()≥, ()
which contradicts assumption (A).
Theorem . Let H be defined as in(),let f(x),ui(x),i= , . . . ,m and vj(x),j= , . . . ,l be three times continuously differentiable functions,let assumptions(A)-(A)hold,and let ξi(x),i= , . . . ,m andηj(x),j= , . . . ,l be twice times continuously differentiable functions. Then for almost all w()∈X×Rm
++×Rl,there exists a Ccurve(w(s),λ(s))of dimension
such that
Hw(s),w(),λ(s)= , w(),λ()=w(), . ()
Proof By Lemma ., there must be aC curve (w(s),λ(s)) of dimension (denoted by w()) such that
Hw(s),w(),λ(s)= , w(),λ()=w(), .
By the classification theorem of one-dimensional smooth manifolds, w() is diffeo-morphic either to a unit circle or to a unit interval. For any w() ∈X× Rm
++ × Rl, ∂Hw()(w(), )/∂wis nonsingular, so
w() cannot be diffeomorphic to a unit circle. That is,w()is diffeomorphic to a unit interval.
Let (w∗,λ∗) be a limit point ofw(). If (w∗,λ∗)∈X×Rm
++×Rl×(, ), because is a regular value ofHw(), (w∗,λ∗)∈H–
w()(), and the Jacobian matrix ofHat (w∗,λ∗) is of full row rank, then by the implicit function theorem,w() can be extended at (w∗,λ∗). This result contradicts the fact that (w∗,λ∗) is a limit point ofw().
Let (w∗,λ∗) = (x∗,y∗,z∗,λ∗). Thus, (w∗,λ∗)∈∂(X×Rm
+ ×Rl×(, ]) and the following three cases are possible:
(a) (w∗,λ∗) = (x∗,y∗,z∗,λ∗)∈X×Rm+ ×Rl× {}. (b) (w∗,λ∗) = (x∗,y∗,z∗,λ∗)∈X×Rm
++×Rl× {}.
(c) (w∗,λ∗) = (x∗,y∗,z∗,λ∗)∈∂(X×Rm
+ ×Rl)×(, ].
Since the equationH(w,w(), ) = has a unique solution (w(), ) inX×Rm
++×Rl× {}, case (b) is impossible.
By the homotopy equation (), we have
( –λk)
∇fx(k)+∇ux(k)y(k)+λkξ
x(k)y(k)+λk
x(k)–x()
+∇vx(k)+λk
ηx(k)–∇vx(k)z(k)= , ()
vx(k)= , ()
Y(k)ux(k)–λkY()u
x()= . ()
Let
I(x) =
i= , . . . ,m: lim k→∞y
(k)
i = +∞
, J(x) =
j= , . . . ,l: lim k→∞z
(k)
j = +∞
.
IfJ(x)=∅, then
( –λk)
∇fx(k)+
i∈I/(x)
∇ui
x(k)y(ik)+λkξi
x(k)y(ik)
+λk
x(k)–x()+∇vx(k)+λ
k
ηx(k)–∇vx(k)z(k)
+ ( –λk)
i∈I(x)
∇ui
x(k)y(ik)+λkξi
x(k)y(ik)= . ()
In case (c), first, we prove thaty∗∈/∂Rm
+. Ify∗∈∂Rm+, then there existi∈ {, . . . ,m}and a sequence of points{(w(k),λk)} ⊂w()such thaty(ik)
→y
∗
i= ask→+∞. From (), we have
y(ik)ui
x(k)=λky()iui
x(). ()
BecauseXand (, ] are bounded, whenk→+∞, the left-hand side of () tends to .
At the same time, the right-hand side of () tends toλ∗y()iui(x
()), which is strictly less
than . This fact results in a contradiction.
Then we only need to prove that the remainder of case (c) is impossible. If not, then there exists a sequence of points{(w(k),λ
k)} ⊂w()such thatui(x(k))→ for somei∈ {, . . . ,m}
ask→+∞. From (), we obtain y(k) →+∞. BecauseXand (, ] are bounded, there exists a subsequence of points (denoted also by{(w(k),λk)}) such thatx(k)→x∗, y(k) →
+∞,z(k)→z∗, andλ
k→λ∗ask→+∞.
Whenλ∗> , from (), the active index set isB(x∗) =I(x∗). Whenλ∗= , the index set
I(x∗)⊂B(x∗).
() Whenλ∗= , from (), by the fact thaty(ik) is bounded for eachi∈/I(x∗), assump-tions (A)-(A), we conclude thatlimk→+∞( –λk)y(ik)= andlimk→+∞( –λk)(yi(k))=y∗i.
Therefore, whenk→+∞, () becomes
ηx∗z∗+
i∈I(x∗) ξi
x(k)y∗i +x∗=x(), ()
which contradicts assumption (A). () When <λ∗< , rewrite () as
i∈I(x)
∇ui
x(k)y(ik)+λkξi
x(k)y(ik)+ λk –λk
x(k)–x()
= –∇fx(k)–
i∈I/(x)
∇ui
x(k)y(ik)+λkξi
x(k)y(ik)
– –λk
∇vx(k)+λk
ηx(k)–∇vx(k)z(k). ()
Lety(Ik)be a vector with components (yi(k)),i∈I(x∗). Then set
ρi(k)=(y (k)
i ) y(Ik) , i∈I
x∗. ()
Note that ≤ρi(k)≤; then there exists a subsequence of{ρi(k)}, still denoted by{ρi(k)}, such thatρi(k)→ρi∗for eachi∈I(x∗) ask→+∞. Furthermore, the vector with componentsρi∗,
i∈I(x∗) is denoted byρ∗; thus, ρ∗ = . Dividing both sides of () by y(Ik) and letting
k→+∞, we have
i∈I(x∗) λ∗iρi∗ξi
x∗= ,
which contradicts assumption (A).
From the discussion above, we conclude that (a) is the only possible case. Hence, the
x-component ofw∗is a K-K-T point of ().
3 Algorithmic analysis
For almost allw()= (x(),y(), )∈X×Rm
++×Rl, by Theorem ., the homotopy gener-ates aCcurve
w(), by differentiating the first equation in (), we obtain the following
theorem.
Theorem . The homotopy pathw()is determined by the following initial value problem
to the ordinary differential equation:
DHw()
w(s),λ(s)
˙
w(s) ˙ λ(s)
= , w(),λ()=w(), , ()
where s is the arc length of the curvew().
As for how to trace numerically the homotopy path, there have been many predictor-corrector algorithms, see [],etc.for reference. Hence we omit them in this paper. In the implementation of the algorithm, generally we need to be devoted to finding the positive direction of the tangent vector at a point onw()which keeps the sign of the determinant DHw()(w,λ)
pT invariant. On the first iteration, the sign is determined by the following lemma. Lemma . Ifw() is smooth,then the positive direction p()at the initial point(w(), )
satisfiessignDHw()(w(),)
p()T
= (–)m+l+.
Proof Let
a= –∇fx()–∇ux()+ ( – λ)ξx()y(), b= –Y()gx(),
then
∂Hw()(w(), ) ∂(w,λ) =
⎛ ⎜ ⎝
I η(x()) a
∇v(x())T Y()∇u(x())T diag(u(x())) b ⎞ ⎟
⎠= (M,M),
where M∈R(n+m+l)×(n+m+l),M∈R(n+m+l)×. The tangent vectorp() ofw() at (w(), ) satisfies
(M,M)p()= (M,M)
p() p()
= , ()
wherep() ∈Rn+m+l,p() ∈R. It is easy to getp() = –M–
Mp() , then
DHw()(w(), )
p()T =
M M
p() T p() T
=
M M
–MT
M–T p
()
=|M|p()
=
I η(x())
∇v(x())T Y()∇u(x())T diag(u(x()))
p() +MTM–TM– M
=diagux()
I η(x()) ∇v(x())T
p
()
+MTM–TM– M
= (–)ldiagux()∇vx()Tηx()p() +MTM–TM– M
.
Sinceu(x()) < , ( +MT
M–TM– M) > , andp() < , so the sign of
DHw()(w(), )
p()T
is (–)m+l+.
The following pseudocode describes the basic steps of a generic predictor-corrector method.
Algorithm .(Euler-Newton method)
Step . Provide an initial guess (w(), ), an initial step lengthh
> , and three small positive numbers> ,> , and> . Setk= .
Step . Compute the directionθ(k)of the predictor step.
(a) Compute a unit tangent vectorp(k).
(b) Determine the directionθ(k)of the predictor step as follows:
If the sign of the determinantDHw()(w
(k),λ
k)
p(k)T
is(–)m+l+, thenθ(k)=p(k).
If the sign of the determinantDHw()(w(k),λk)
p(k)T
is(–)m+l, thenθ(k)= –p(k). Step . Compute a corrector point (w(k+),λ
k+).
¯
w(k),λ¯k
=w(k),λk
+hkθ(k),
w(k+),λk+
=w¯(k),λ¯k
–DHw()
¯
w(k),λ¯k
+
Hw()
¯
w(k),λ¯k
.
If Hw()(w(k+),λ
k+) ≤, then lethk+=min{h, hk}, and go to Step .
If Hw()(w(k+),λ
k+) ∈(,), then lethk+=hk, and go to Step .
If Hw()(w(k+),λ
k+) ≥, then lethk+=max{–h, (hk/)},k=k+ , and go to Step .
Step . Ifλk+≤, then stop. Otherwise,k=k+ , and go to Step .
4 Conclusions
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MS carried out the main work and drafted the manuscript. JL participated in completing the proof of Lemma 2.3. ZX participated in completing the proof of Theorem 2.1. All authors read and approved the final manuscript.
Author details
1School of Mathematical Sciences, Luoyang Normal University, Luoyang, 471022, P.R. China.2National Key Laboratory of
Science and Technology on Advanced Composites in Special Environments, Harbin Institute of Technology, Harbin, 150080, P.R. China.
Acknowledgements
This work was supported by NSFC-Union Science Foundation of Henan (No. U1304103) and Natural Science Foundation of Henan Province (No. 122300410261).
Received: 1 June 2014 Accepted: 14 July 2014 Published:15 Aug 2014
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