An interior algorithm for nonlinear optimization that combines line search and trust region steps.

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An Interior Algorithm for Nonlinear Optimization That
Combines Line Search and Trust Region Steps
R.A. Waltz∗
J.L. Morales†
J. Nocedal∗
D. Orban∗
September 8, 2004
Abstract
An interior-point method for nonlinear programming is presented. It enjoys the
flexibility of switching between a line search method that computes steps by factoring
the primal-dual equations and a trust region method that uses a conjugate gradient
iteration. Steps computed by direct factorization are always tried first, but if they are
deemed ineffective, a trust region iteration that guarantees progress toward stationar-
ity is invoked. To demonstrate its effectiveness, the algorithm is implemented in the
Knitro [6, 28] software package and is extensively tested on a wide selection of test
problems.
1Introduction
In this paper we describe an interior method for nonlinear programming and discuss its
software implementation and numerical performance. A typical iteration computes a pri-
mary step by solving the primal-dual equations (using direct linear algebra) and performs a
line search to ensure decrease in a merit function. In order to obtain global convergence in
the presence of nonconvexity and Hessian or Jacobian rank deficiencies, the primary step is
replaced, under certain circumstances, by a safeguarding trust region step. The algorithm
can use exact second derivatives of the objective function and constraints or quasi-Newton
approximations.
The motivation for this paper is to develop a new interior point algorithm, implemented
in the Knitro software package [28], which is more robust and efficient than either a pure
trust region or a pure line search interior approach. The algorithm implemented in the first
release of Knitro [6] is a trust region method that uses a null-space decomposition and
a projected conjugate gradient iteration to compute the step. This iterative approach has
the advantage that the Hessian of the Lagrangian need not be formed or factored, which is
∗Department of Electrical and Computer Engineering, Northwestern University. These authors were
supported by National Science Foundation grants CCR-9987818, ATM-0086579, and CCR-0219438 and
Department of Energy grant DE-FG02-87ER25047-A004.
†Departamento de Matem´ aticas, ITAM, M´ exico. This author was supported by Asociaci´ on Mexicana de
Cultura, A.C. and CONACyT grant 39372-A.
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effective for many large problems. The disadvantage is that the projected conjugate gradient
iteration can be expensive when the Hessian of the Lagrangian of the nonlinear program
is ill-conditioned, in which case it is preferable to use direct linear algebra techniques to
compute the step.
By designing the algorithm so that it computes steps using direct linear algebra when-
ever the quality of these steps can be guaranteed, and falling back on a trust region step
otherwise, we can easily implement the new method within the existing Knitro package.
Moreover, since the algorithm can reduce to a pure line search or a pure trust region ap-
proach, we are able to implement two different interior methods in a unified algorithmic
and software framework.
The problem under consideration has the form
min
x
s.t.
f(x)(1.1a)
h(x) = 0(1.1b)
g(x) ≤ 0,(1.1c)
where f : Rn→ R, h : Rn→ Rland g : Rn→ Rmare twice continuously differentiable
functions. A variety of line search interior algorithms have been proposed [12, 16, 25, 27, 30],
several of which have been implemented in high-quality software packages [2, 25, 27]. The
search direction is computed in these algorithms by factoring the primal-dual system. In
order to achieve robustness, these line search approaches must successfully address two
issues:
• How to define the search direction when the quadratic model used by the algorithm
is not convex;
• How to handle rank deficiency (and near deficiency) of the Hessian of the Lagrangian
and constraint Jacobian.
The first issue is often addressed by adding a multiple of the identity matrix to the Hessian of
the Lagrangian in order to convexify the model. This strategy was first shown to be effective
in the context of nonlinear interior methods by the Loqo software package [25]. The second
issue is handled differently in each software implementation. The difficulties caused by
rank deficient constraint Jacobians are sometimes addressed at the linear algebra level by
introducing perturbations during the factorization of the KKT matrix [1]. Other approaches
include the use of ℓ1 or ℓ2 penalizations of the constraints, which provide regularization
[16, 24], and the use of a feasibility restoration phase [15, 27].
In this paper we describe a mechanism for stabilizing the line search iteration that is
different from those proposed in the literature. It consists of falling back, under certain
conditions, on a trust region step that is guaranteed to make progress toward feasibility
and optimality. A challenge is to design the algorithm so that there is a smooth transition
between line search and trust region steps. We will argue that the algorithm presented in this
paper is not more expensive than other approaches, has favorable convergence properties,
and performs well on standard test problems.
Notation. Throughout the paper ? · ? denotes the Euclidean norm.
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2Outline of the Algorithm
We consider an interior method that replaces the nonlinear program (1.1) by a sequence of
barrier subproblems of the form
min
z
ϕµ(z) ≡ f(x) − µ
m
?
i=1
lnsi
(2.1a)
s.t.h(x) = 0(2.1b)
g(x) + s = 0.(2.1c)
Here s > 0 is a vector of slack variables, z = (x,s), and µ > 0 is the barrier parameter. The
Lagrangian function associated with (2.1) is defined by
L(z,λ;µ) = ϕµ(z) + λT
hh(x) + λT
g(g(x) + s),(2.2)
where λh∈ Rland λg ∈ Rmare Lagrange multipliers and λ = (λh,λg). The first-order
optimality conditions for the barrier problem (2.1) can be written as
?
∇f(x) + Ah(x)Tλh+ Ag(x)Tλg
SΛge − µe
?
=
?
0
0
?
,(2.3)
together with (2.1b) and (2.1c) and the restriction that s and λgbe nonnegative. Here S
and Λgdenote diagonal matrices whose diagonal entries are given by the vectors s and λg,
respectively, and Ahand Agare the Jacobian matrices of h and g.
Applying Newton’s method to the system (2.3), (2.1b), (2.1c), from the current iterate
(z,λ) results in the primal-dual system
?
W(z,λ;µ)
A(x)
A(x)T
0
??
dz
dλ
?
= −
?
∇zL(z,λ;µ)
c(z)
?
,(2.4)
where we have defined
dz=
?
dx
ds
?
,dλ=
?
dh
dg
?
,c(z) =
?
h(x)
g(x) + s
?
(2.5)
and
A(x) =
?
Ah(x)
Ag(x)
0
I
?
,W(z,λ;µ) =
?
∇2
xxL(z,λ;µ)
0
0
S−1Λg
?
. (2.6)
The new iterate is given by
z+= z + αzdz,λ+= λ + αλdλ.(2.7)
The steplengths αzand αλare computed in two stages. First we compute
αmax
z
αmax
λ
=max{α ∈ (0,1] : s + αds≥ (1 − τ)s}
max{α ∈ (0,1] : λg+ αdg≥ (1 − τ)λg},
(2.8a)
=(2.8b)
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with 0 < τ < 1 and typically close to one. (In our tests we use τ = .995.) The algorithm
then performs a backtracking line search that computes steplengths
αz∈ (0,αmax
z
],αλ∈ (0,αmax
λ ],(2.9)
providing sufficient decrease in a merit function (to be defined below). The procedure for
updating the barrier parameter µ is described in Section 3.5.
The iteration (2.4)-(2.7) provides the basis for most line search interior methods. This
remarkably simple approach must, however, be modified to cope with non-convexity and
to prevent convergence to non-stationary points [7, 26]. Instead of modifying the primal-
dual matrix, as is commonly done, we use a safeguarding trust region step to stabilize the
iteration, for two reasons. First, when W is not positive definite on the null-space of A
(the negative curvature case), adding to W a multiple of the identity matrix may introduce
undesirable distortions in the model and can also require several factorizations of the primal-
dual system. In the negative curvature case, we prefer to compute a descent direction using
a null space approach in which the tangential component of the step is obtained by using
a projected Krylov iteration, as is done with conjugate gradients in the Knitro package
[28], or using a Lanczos method, as is done in the GALAHAD package [19].
Second, we would like to take advantage of the robustness of trust region steps in the
presence of Hessian or Jacobian rank deficiencies. We have in mind trust region methods
that provide Cauchy decrease for both feasibility and optimality at every iteration. Since it
is known that, when line search iterations converge to non-stationary points, the steplengths
αzor αλin (2.7) converge to zero, we monitor these steplengths. If they become smaller
than a given threshold, we discard the line search iteration (2.4)-(2.7) and replace it with a
trust region step. The resulting algorithm possesses global convergence properties similar
to those of the algorithms implemented in FilterSQP [15] and Knitro [5].
We outline the method in Algorithm 2.1. There φν(z) denotes a merit function using
a penalty parameter ν, and Dφν(z;dz) denotes the directional derivative of φν along a
direction dz. An inertia-revealing symmetric indefinite factorization [4] of the primal-dual
matrix in (2.4) provides its number of negative eigenvalues. If this number exceeds l + m,
then dzcannot be guaranteed to be a descent direction (see, e.g., [21, Lemma 16.3]), and
the primal-dual step is discarded.
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Algorithm 2.1 Outline of the Interior Algorithm
Choose z0 = (x0,s0) and the parameters 0 < η, 0 < δ < 1 and an integer imax. Com-
pute initial values for the multipliers λ0, the trust-region radius ∆0> 0, and the barrier
parameter µ0> 0. Set k = 0.
Repeat until a stopping test for the nonlinear program (1.1) is satisfied
Repeat until a termination test for the barrier problem (2.1) is satisfied
Factor the primal-dual system (2.4) and record the number neig
of negative eigenvalues of its coefficient matrix.
Set LineSearch = False.
If neig ≤ l + m
Solve (2.4) to obtain the search direction d = (dz,dλ).
Compute αmax
z ,αmax
λ
using (2.8a),(2.8b).
If min{αmax
z ,αmax
Set j = 0, αT= 1.
Repeat while (j ≤ imax), (αT> δ) and LineSearch == False
If φν(zk+ αTαmax
z dz) ≤ φν(zk) + ηαTαmax
Set αz= αTαmax
z , αλ= αTαmax
Set zk+1,λk+1using (2.7).
Compute ∆k+1and set LineSearch = True.
Else
Set j ← j + 1.
Choose a smaller value of αT.
Endif
End Repeat
Endif
Endif
If LineSearch == False
Compute (zk+1,λk+1) using a globally convergent safeguarding
trust region method.
Compute ∆k+1.
Endif
Set µk+1← µk.
Set k ← k + 1.
End Repeat
Reset the barrier parameter µkso that µk< µk−1.
End Repeat
λ} > δ
z Dφν(zk;dz)
λ.
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