A filter SQP algorithm without a feasibility restoration phase

A filter SQP algorithm without a feasibility restoration phase^*

Chungen Shen^I,II; Wenjuan Xue^III; Dingguo Pu^I

^IDepartment of Mathematics, Tongji University, China, 200092

^IIDepartment of Applied Mathematics, Shanghai Finance University, China, 201209

^IIIDepartment of Mathematics and Physics, Shanghai University of Electric Power, China, 200090 E-mail: [e-mail]

ABSTRACT

In this paper we present a filter sequential quadratic programming (SQP) algorithm for solving constrained optimization problems. This algorithm is based on the modified quadratic programming (QP) subproblem proposed by Burke and Han, and it can avoid the infeasibility of the QP subproblem at each iteration. Compared with other filter SQP algorithms, our algorithm does not require any restoration phase procedure which may spend a large amount of computation. We underline that global convergence is derived without assuming any constraint qualifications. Preliminary numerical results are reported.

Mathematical subject classification: 65K05, 90C30.

Key words: filter, SQP, constrained optimization, restoration phase.

1 Introduction

In this paper, we consider the constrained optimization problem:

where c(x) = (c₁(x), c₂(x), ... , c_m1 (x))^T , c(x) = (c_m1+1(x), c_m1+2(x), ... , c_m(x))^T , ={1, 2, ... , m₁}, = {m₁ + 1, m₁ + 2, ... , m}, ƒ: ⁿ

and c_i: ⁿ

, i

are twice continuously differentiable functions. The feasible set of the problem (NLP) is denoted by X = {x

Rⁿ | c(x) = 0, c(x) < 0}.

The sequential quadratic programming (SQP) method has been widely used for solving the problem (NLP). It generates a sequence {x_k} converging to the desired solution by solving the quadratic programming (QP) subproblem:

where ρ> 0 is the trust region radius, B_k

^n×n is a symmetric positive definite matrix. At each iteration x_k, the solution d of the QP subproblem is regardedas a trial step and the next trial iterate has the form x_k + d. The acceptance of this trial step depends on whether the trial iterate x_k+ d makes some merit function descent. Generally, this merit function is a type of penalty function with some parameters whose adjustment can be problematic. Fletcher and Leyffer [7] proposed a trust region filter SQP method to solve the problem (NLP) instead of traditional merit function SQP methods. In addition, the computational results presented in Fletcher and Leyffer [7] are also very encouraging.

Recently, this topic got high importance in recent years (see [6, 17, 18, 22, 23]). Trust region filter SQP methods have been studied by Fletcher, Leyffer and Toint in [8] and by Fletcher, Gould, Leyffer, Toint and Wächter in [9]. In this latter paper, an approximate solution of the QP subproblem is computed and the trial step is decomposed into normal and tangential components. Gonzaga, Karas and Vanti [10] presented a general framework for filter methods where each step is composed of a feasibility phase and an optimality phase. Similar filter method was proposed by Ribeiro, Karas and Gonzaga in [21]. In all these papers only the global convergence of the proposed methods is analyzed. On the other hand, in [24], Ulbrich studied the local convergence of a trust region filter SQP method. Anyway, the components in the filter adopted in [24], differs from those in [7, 8, 9]. It should be underlined that the filters approach has been used also in conjunction with the line search strategy (see Wächter and Biegler [26, 27]); with interior point methods (see Benson, Shanno and Vanderbei [5]; Ulbrich, Ulbrich and Vicente [25]) and with the pattern search method (see Audet and Dennis [1]; Karas, Ribeiro, Sagastizábal and Solodov [15]).

However, all filter algorithms mentioned above include the provision for a feasibility restoration phase if the QP subproblem becomes inconsistent. Although any method (e.g. [3, 4, 16]) for solving a nonlinear algebraic system of equalities and inequalities can be used to implement this calculation, a large amount of computation may be spent. In this paper, we incorporate the filter technique with the modified QP subproblem for solving general constrained optimization problems. The main feature of this paper is that there is no restoration phase procedure. Another feature is that global convergence of our algorithm is established without assuming any constraint qualifications.

This paper is organized as follows. In section 2, we describe how the modified QP subproblem is embedded in the filter algorithm. In section 3, it is proved that the algorithm is well defined, and is globally convergent to a stationary point. If Mangasarian-Fromowitz constraint qualification (MFCQ) holds at this stationary point, then it is a KKT point. In section 4, we report some numerical results.

2 Algorithm

It is well-known that the Karush-Kuhn-Tucker (KKT) conditions for the problem (NLP) are:

where

is the Lagrangian function, µ

R^m1 and λ R^m-m1 are the multipliers corresponding to the constraints of the problem (NLP).

The SQP method generates iterates by solving the subproblem (1.1). However, the subproblem (1.1) may be inconsistent if xk is not a feasible point for the problem (NLP). In order to avoid this bad situation, we solve a modified QP subproblem instead of the subproblem (1.1). Before introducing the modified QP subproblem, we first solve the problem

where the scalar σ_k > 0 is used to restrict d in norm. Let _k denote the solution of (2.4). If σ_k < ρ and q_l(_k ) = 0, then the QP subproblem (1.1) is consistent. The problem (2.4) can be reformulated as a linear program

where vectors e_m1 = (1, 1, ... , 1)^T

^m1 and e_m-m1 = (1, 1, ... , 1)^T

^m-m1.

Now we define that (x_k, σ_k ) and (x_k,σ_k ) are equal to 2 and c(x_k)^T

_k + c(x_k), respectively, where (_k , ₁, ₂) denotes the solution of (2.5). At each iteration, we generate the trial step by solving the modified QP subproblem

Here, at each iterate, we require that ρ_k be greater than σ_k . So, the choice of σ_k depends on ρ_k.

Define Φ(x_k,σ_k ) = , where

_i (x_k,σ_k ) and _i (x_k,σ_k ) denote the ith component of (x_k,σ_k ) and (x_k,σ_k ) respectively. Actually, Φ(x_k,σ_k ) = q_l(_k ) with the scalar σ_k > 0. Obviously, d = 0 is feasible for the problem (2.4) and then q_l(_k ) < q_l(0) = V (x_k), where the constraint violation V (x) is defined as

Therefore, Φ(x_k,σ_k ) < V (x_k).

Let d denote the solution of QP(x_k, B_k, σ_k, ρ_k ). Let the trial point be = x_k+ d. The KKT conditions for the subproblem QP(x_k, B_k, σ_k, ρ_k ) are

where e_n= (1, 1, ... , 1)^T

ⁿ, µk

^m1, λk

^m-m1, λ^l_k

ⁿ and λ^u_k

ⁿ.

In our filter, we consider pairs of values (V (x), ƒ(x)). Definitions 1-4 below are very similar to those in [7, 8].

Definition 1. The iterate x_kdominates the iterate x_lif and only if V (x_k) < V (x_l) and ƒ(x_k) < ƒ(x_l). And it is denoted by x_k< x_l.

Thus, if x_k< x_l, the latter is of no real interest to us since x_kis at least as good as x_lwith respect to the objective function's value and the constraint violation. Furthermore, if x_k< x_l, we say that the pair (V (x_k), ƒ(x_k)) dominates the pair (V (x_l), ƒ(x_l)).

Definition 2. The kth filter is a list of pairs {(V (x_l), ƒ(x_l))}_l<k , such that nopair dominates any other.

Let

_k denote the indices in the kth filter, i.e.,

Filter methods accept a trial point = x_k+ d if its corresponding pair (V (), ƒ()) is not dominated by any other pair in the kth filter, neither the pair corresponding to x_k, i.e., (V (x_k), ƒ (x_k)).

In a filter algorithm, one accepts a new pair (V (), ƒ()) if it cannot be dominated by other pairs in the current filter. Although the definition of filter is simple, it needs to be refined a little in order to guarantee the global convergence.

Definition 3. A new trial point is said to be "acceptable to x_l", if

is satisfied, where γ_i (0, ½), i = 1, 2 are two scalars.

Definition 4. A new trial point is said to be "acceptable to the kth filter" if is acceptable to x_lfor all l

_k .

Therefore, to accept a new pair into the current filter, we should test the conditions defined in Definition 4. If the new trial point is acceptable in the sense of Definitions 3 and 4, we may wish to add the corresponding pair to the filter. Meanwhile, any pair in the current filter dominated by the new pair is removed.

In order to restrict V (x_k), it still needs an upper bound condition for accepting a point. The trial point satisfies the upper bound condition if

holds, where

_k is a positive scalar. Here _k is updated at each iteration and it may converge to zero for some instances. We aim to control the constraint violation by setting _k . In current iteration k, if a trial step is accepted and Φ(x_k,σ_k ) = 0, then we will keep _k unchanged in next iteration. Otherwise we will reduce _k in next iteration. The detailed information about the update of _k is given in Algorithm 2.1.

Denote

as the quadratic reduction of ƒ(x), and

as the actual reduction of ƒ(x). In our algorithm, if a trial point x_k+ d is acceptable to the current filter and x_k, and

the sufficient reduction criterion of the objective function ƒ should be satisfied, i.e.,

Now we are in the position to state our filter SQP algorithm.

Algorithm 2.1.

Step 1. Initialization.

Given initial point x₀, parameters η (0, ½), γ₁ (0, ½), γ₂ (0, ½), γ (0, 0.1), r (0, 1), ₀ > 0, ρ_min > 0, > ρ_min, ρ [ρ_min, ], σ₀γρ, ρ). Set k = 0, Flag=1, put k into _k .

Step 2. Compute the search direction.

Compute the search direction d from the subproblem QP(x_k, B_k, σ_k, ρ). If ρ > ρ_min, set d_s = d and Φ_s = Φ(x_k, σ_k).

Step 3. Check for termination.

If d = 0 and V (x_k) = 0, then stop;

Else if V (x_k) > 0 and Φ(x_k, σ_k ) - V (x_k) = 0, then stop.

Step 4. Test to accept the trial point.

4.1. Check acceptability to the filter.

If the upper bound condition (2.8) holds for = x_k+ d, Φ(x_k, σ_k) = 0, and x_k+ d is acceptable to both the kth filter and x_k, then go to 4.3.

Otherwise, if Φ(x_k, σ_k) = 0, then set ρ = ½ ρ and choose σ_k (γρ, ρ), go to Step 2;

4.2. Compute t_k, the first number t of the sequence {1, r, r² , ... , } satisfying

Set Flag=0, d = d_s, and go to 4.4.

4.3. Check for sufficient reduction criterion.

If Δ ƒ(d) < ηΔq(d) and Δq(d) > 0, then ρ = ½ ρ and choose σ_k (γρ, ρ), and go to Step 2;

otherwise go to 4.4.

4.4. Accept the trial point.

Set

and

Step 5. Augment the filter if necessary.

If Flag=1 and Δq(d_k) < 0, then include (V (x_k), ƒ(x_k)) in the kth filter.

If Flag=0, then set

_{k +1} = V (x_k+ t_kd_k); otherwise, set _{k +1} = _k and leave the filter unchanged.

Step 6. Update.

Compute B_{k +1}. Set σ_{k +1} (γρ, ρ) and k := k + 1, Flag=1, go to Step 2.

Remark 1. The role d_sin Step 2 is to record a trial step d with ρ > ρ_min. By Step 4.1, we know that if Φ(x_k,σ_k ) > 0, then both conditions in Step 4.1 are violated. A trial iterate x_k+ d with Φ(x_k, σ_k ) > 0 may be considered as a "worse" step which may be far away from the feasible region. So Step 4.2 is executed to reduce the constraint violation. The step d_sis used in Step 4.2 due to the descent property of d_sproved by Lemma 3.4. In addition, it should be interpreted that, at the beginning of iteration k, the pair (V (x_k), ƒ(x_k)) is not in the current filter, but x_kmust be acceptable to the current filter.

Remark 2. Our algorithm has three loops: the loop 2-(4.1)-2, the loop 2-(4.3)-2, and the loop 2-6-2. We observed that the radius ρ is allowed to reduce to a value less than ρ_min in the loops 2-(4.1)-2 and 2-(4.3)-2. But in the loop 2-6-2 the radius ρ is not less than ρ_min. At each iteration, the first trial radius ρ is greater than or equal to ρ_min. Subsequently, trial radius ρ may not stop decreasing until either the filter acceptance criteria are satisfied or Step 4.2 is executed. Hence, ρ may be less than ρ_min during the execution process of the loops 2-(4.1)-2 and 2-(4.3)-2.

As for proving global convergence, we use the terminology firstly introduced by Fletcher, Leyffer and Toint [8]. We call d an ƒ-type step if Δq(d) > 0, indicating that then the sufficient reduction criterion (2.12) is required. If d is accepted as the final step d_kin the kth iteration, we refer to k as an ƒ-type iteration.

Similarly, we call d a V -type step if Δq(d) < 0. If d is accepted as the final step d_kin iteration k, we refer to k as a V -type iteration. In addition, if x_kis generated by Step 4.2, we also refer to it as a V -type iteration.

If ƒ (x_{k +1}) < ƒ (x_k), then we regard the step d_kas an ƒ monotone step. Obviously, an ƒ -type step must be an ƒ monotone step.

3 Global convergence

In this section, we prove the global convergence of Algorithm 2.1. Firstly, we give some assumptions:

(A1) Let {x_k} be generated by Algorithm 2.1 and {x_k}, {x_k+ d_k} are contained in a closed and bounded set S of ⁿ;

(A2) All the functions ƒ , c_i , i

are twice continuously differentiable on S.

(A3) The matrix B_kis uniformly positive definite and bounded for all k.

Remark 3. Assumption (A1) is reasonable. It may be forced if, for example, the original problem involves a bounded box among its constraints.

Remark 4. A consequence of Assumption (A3) is that there exist constants δ, M > 0, independent of k such that δy

² < y^T B_ky < M

y

² for all y

ⁿ. Assumptions (A1) and (A2) imply boundedness of

²c_i(x)i

, and

² ƒ (x) on S. Without loss of generality, we may assume

²c_i(x)< M, i

,

²ƒ (x)< M, x

S.

Lemma 3.1. Assume

is not a stationary point of V in the sense that 0

V (), where

V denotes the Clarke subdifferential of V . Then there exists a scalar > 0 and a neighborhood N() of , such that Φ(x, σ) - V(x) < - for all σ > γρ_minand all x

N ().

Proof. By [2, Lemma 2.1], 0

V () implies Φ(, γρ_min) - V () < 0, where γ and ρ_min are from Algorithm 2.1. By the continuity of the function Φ (· ,γρ_min) - V (·) on ⁿ, there exists a neighborhood N() and a scalar > 0 such that Φ(x,γρ_min) - V(x) < - whenever x

N (). The condition σ > γρ_min together with the definition of Φ yields Φ (x,σ) - V (x) < Φ(x, γρ_min) - V (x). Therefore, Φ(x, σ) - V (x) < - holds for all σ > γρ_min and all x

N ().

Remark 5. It can be seen that depends on such that 0

V () (i.e., = ()), because it must satisfy Φ(,σ) - V ()< -.

Lemma 3.2. Let d_k= 0 be a feasible point of QP (x_k, B_k, σ_k, ρ_k ). Then x_k is a stationary point of V (x). Moreover, if x_k

X, then x_k is a KKT point of the problem (NLP).

Proof. Since d_k= 0 implies Φ(x_k,σ_k ) - V (x_k) = 0, it follows from [2, Lemma 2.1] that x_kis a stationary point of V (x). If V (x_k) = 0 and d_k= 0, then it follows from [2, Lemma 2.2] that x_kis a KKT point for the problem (NLP).

Lemma 3.3. Let Assumptions (A1)-(A3) hold and d be a feasible point of the subproblem QP(x, B, σ, ρ), then we have

for t [0, 1].

Proof. By Taylor Expansion formula, the feasibility of d and Assumption (A2), we have that for i

and t [0, 1],

and for i

and t [0, 1],

where the vector z_iis between x and x + td. The term ½t²nMρ² in (3.15) and (3.16) is derived because

where we use that ·² < n·². Formulae (3.15) and (3.16) combining with definitions of V (x) and Φ(x,σ) yield (3.14).

The following lemma shows that the loop in Step 4.2 is finite.

Lemma 3.4. Let Assumptions (A1)-(A3) hold, η (0, ½) and

satisfies 0

V (). Then, there exist a scalar > 0 and a neighborhood N () of

such that for any x

N () and any d feasible for the problem QP(x, B, σ, ρ) with γρ_min< σ < ρ < , it holds that

for all t (0, ].

Proof. It follows from Lemma 3.1 that there exists a neighborhood N () and () > 0 such that Φ(x, σ) - V(x) < -() whenever x

N (). Combining this with (3.14), we have

Hence, the inequality (3.17) holds for all t (0, ()], where

Hence, if 0

V (x_k), (2.13) follows taking = x = x_kand d = d_s. It implies that the loop in Step 4.2 is finite.

The following lemma shows that the iterate sequence {x_k} approaches a stationary point of V (x) when Step 4.2 of Algorithm 2.1 is invoked infinitely many times.

Lemma 3.5 If Step 4.2 of Algorithm 2.1 is invoked infinitely many times, then there exists an accumulation point

of {x_k} such that the sequence {_k } converges to V (), where 0

V (), i.e., is a stationary point of V (x).

Proof. Since Step 4.2 is invoked infinitely many times. By Step 5 _{k +1} is reset by V (x_k+ t_kd_k) infinitely. From Step 2 and 4.2, we note that if Step 4.2 is invoked at iteration k, the radius ρ in QP(x_k, B_k, σ_k, ρ) associated with d_sis greater than or equal to ρ_min. Define ={k | _{k +1} is reset by V (x_k+ t_kd_k)}. Obviously, is an infinite set. The inequality (2.13) ensures V (x_{k +1}) = _{k +1} for all k

. The upper bound condition (2.8) ensures V (x_{k +1}) < _k = _{k +1} for all k

. Therefore, V (x_k) < _k for all k. This together with (2.13) yields _{k +1}< V (x_k) < _k for all k

. For k

, _{k +1} = _k . Therefore, {_k } is a monotonically decreasing sequence and also has a lower bound zero. Then there exists an accumulation point of {x_k} such that {_{k +1}}

V (). If 0

V (), by Lemma 3.1, there exists a neighborhood N () of and () > 0, such that

whenever x_k

N (). This together with (2.13) yields

for k

and x_k

N(). By the mechanism of Algorithm 2.1 and Lemma 3.4, we have t_k> r(). It follows with (3.19) that

for k

and x_k

N(). Letting k

tend to infinity in above inequality, the limit in the left-hand side is zero while the limit in the right-hand side is less than zero, which is a contradiction. Therefore 0

V ().

Lemma 3.6. Consider sequences {V (x_k)} and {ƒ (x_k)} such that ƒ(x_k) is monotonically decreasing and bounded below. If for all k, either V (x_k+1)-V (x_k) < -γ₁V (x_k+1) or ƒ(x_k) - ƒ(x_k+1) > γ₂V (x_k+1) holds, where constants γ₁and γ₂are from (2.7a) and (2.7b), then V (x_k) 0, for k +.

Proof. See [8, Lemma 1].

Lemma 3.7. Suppose Assumptions (A1)-(A3) hold. If there exists an infinite sequence of iterates {x_k} on which (V (x_k), ƒ(x_k)) is added to the filter, where V (x_k) > 0 and {ƒ(x_k)} is bounded below, then V (x_k) 0 as k +.

Proof. Since inequalities (2.7a) and (2.7b) are the same as [8, (2.6)], the conclusion follows from Lemma 3.6 and [8, Corollary].

Lemma 3.8. Suppose Assumptions (A1)-(A3) hold. Let d be a feasible point of QP(x_k, B_k, σ_k, ρ). If Φ(x_k, σ_k ) = 0, it then follows that

Proof. The proof of this lemma is very similar to the proof [8, Lemma 3]. By Taylor's theorem, we have

where y denotes some point on the line segment from x_kto x_k+ d. This together with (2.9) and (2.10) implies

Then (3.20) follows from the boundedness of

² ƒ(y) and B_k, and d

< ρ. Since Φ(x_k,σ_k ) = 0, QP(x_k, B_k, σ_k, ρ) reduces to (1.1). For i

, it follows that

where y_idenotes some point on the line segment from x_kto x_k+ d. By feasibility of d,

and

follow in a similar way. It follows with the definition of V (x) that (3.21) holds.

Lemma 3.9. Suppose Assumptions (A1)-(A3) hold. Let d be a feasible point of QP(x_k, B_k, σ_k, ρ) with Φ(x_k, σ_k ) = 0. Then x_k+ d is acceptable to the filter if ρ² < , where .

Proof. It follows from (3.21) that V (x_k+ d) - τ_k < -γ₁V (x_k+ d) holds if ρ² < . By the definition of τ_k, (2.7a) is satisfied. Hence x_k+ d is acceptable to the filter.

In order to prove that the iterate sequence generated from Algorithm 2.1 converges to a KKT point for the problem (NLP), some constraint qualification should be required, such as the well-known MFCQ. Thus we review its definition as follows.

Definition 5 (See [2]). MFCQ is said to be satisfied at x, with respect to the underlying constraint system c(x) = 0, c(x) < 0, if there is a z

ⁿ such that the gradients c_i(x), i

are linearly independent and the following systems

are satisfied.

Proposition 3.10. Suppose that MFCQ is satisfied at x^*

X, then there exists a neighborhood N (x^* ) of x^* such that

1. MFCQ is satisfied at every point in N(x^* ) X;

2. inequality

holds, where the vectors λ_k, µ_k generated by QP(x_k, B_k, σ_k, ρ_k ) are multiplier vectors associated with x_k

N (x^* ).

Proof. The first result is established in [20, Theorem 3]. The second result is established in [2, Theorem 5.1].

If MFCQ does not hold at a feasible x^*, then the second statement of Proposition 3.10 cannot be guaranteed. All feasible points of the problem (NLP) at which MFCQ does not hold will be called non-MFCQ points.

The following lemma shows that (1.1) is consistent when x_kapproaches a feasible point at which MFCQ holds and both the quadratic reduction and the actual reduction of the objective function have sufficient reduction. Its proof is similar to that of [8, Lemma 5].

Lemma 3.11. Suppose Assumptions (A1)-(A3) hold and let x^*

S be a feasible point of problem (NLP) at which MFCQ holds but which is not a KKT point. Then there exists a neighborhood Nº of x^* and positive constants , ν and , such that for all x_k

S

Nº and all ρ for which

it follows that QP(x_k, B_k, σ_k, ρ) with Φ(x_k, σ_k ) = 0 has a feasible solution d at which the predicted reduction satisfies

the sufficient reduction condition (2.12) holds, and the actual reduction satisfies

Proof. Since x^* is a feasible point at which MFCQ holds but it is not a KKT point, it follows that the vectors c_i(x^*), i

are linearly independent, and there exists a vector s^* that satisfies

where

s^* = 1. Let c(x_k)⁺ = c(x_k)^T

c(x_k))^-1

c(x_k)^T and c(x_k) denote the matrix with columns c_i(x_k), i

. It follows from linear independence and continuity that there exists a neighborhood of x^* in which c(x_k)⁺ is bounded. Let P = -c(x_k)^+T c(x_k) and s = (I - c(x_k)c(x_k)⁺) s^* / (I -c(x_k)c(x_k)⁺) s^* if is not empty, otherwise P = 0 and s = s^*. Let p = P. It follows from (3.25) and (3.27) by continuity that there exists a (smaller) neighborhood ^* and a constant > 0 such that

when x_k

^*. By definition of P, it follows that p = O(V (x_k)), and thus we can choose the constant ν in (3.22) sufficiently large so that ρ> p for all x_k

^*.

We now consider the solution of (1.1). The line segment defined by

for a fixed value of ρ > p. Since ρ > p, and P and s are orthogonal, it implies

From (3.29) and the definitions of P, s, d_α satisfies the equality constraints c(x_k) +c(x_k)^Td_α = 0 of (1.1) for all α [0, 1].

If x_k

^*

S and i

\(x^*), then there exists positive constants and , independent of ρ, such that

for all vectors s such that s

< 1, by the continuity of c_i(x_k) and boundedness of c_i(x_k) on S. It follows that

for all vectors d such that Cd

< ρ. Therefore, inactive constraints do not affect the solution of (1.1) if ρ satisfies ρ < /.

For active inequality constraints i

(x^*), it follows from (3.28) and (3.29) that c_i(x_k) +c_i(x_k)^T d₁ = c_i(x_k) +c_i(x_k)^TP +(ρ - p)c_i(x_k)^Ts < c_i(x_k) + c_i(x_k)^TP - (ρ - p) < 0 if ρ > p + (c_i(x_k) +c_i(x_k)^TP)/. We obtain from the definition of P that

Thus we can choose the constant ν in (3.22) sufficiently large so that c_i(x_k) + c_i(x_k)^Td₁< 0, i

(x^*). Therefore d₁ is feasible in (1.1) with respect to the active inequality constraints, and hence to all the constraints by above results. Combining with the fact that (1.1) is equivalent to QP(x_k, B_k, σ_k, ρ) with Φ(x_k, σ_k ) = 0, we obtain that QP(x_k, B_k, σ_k, ρ) with Φ(x_k, σ_k ) = 0 is consistent for all x_k

* and all ρ satisfying (3.22) for any value of < /.

Next we aim to obtain a bound on the predicted reduction Δq(d). We note that q(0) - q(d₁) = - ƒ(x_k)^T (P + (ρ - p)s) - ½ d₁^T B_kd₁. Using (3.28), bounds on B_kand P, and ρ> p = O(V (x_k)), we have

If ρ < , then q(0) - q(d₁) > ½ ρ + O(V (x_k)). Since d₁ is feasible and p = O(V (x_k)), it follows that the predicted reduction (2.9) satisfies

for some sufficiently large and independent of ρ. Hence, (3.23) is satisfied if ρ > 6V (x_k)/. This condition can be achieved by making the constant ν in (3.22) sufficiently large.

Next we aim to prove (3.24). By (3.20) and (3.23), we have

Then, if ρ < (1 - η)/(3nM), it follows that (2.12) holds. By (2.12), (3.21) and (3.23), we have ƒ(x_k) - ƒ(x_k+ d) - γ₂V (x_k+ d) = Δƒ(d) - γ₂V (x_k+ d) > ηρ - ½ γ₂mnρ² M > 0 if ρ < η/(γ₂mnM). Therefore, we may define the constant in (3.22) to be the least of η/(γ₂mnM) and the values (1 - η)/ (3nM), ρ < and /, as required earlier in the proof.

Lemma 3.12. Suppose Assumptions (A1)-(A3) hold and let x^*

S be a feasible point of problem (NLP) at which MFCQ holds but which is not a KKT point. Then there exists a neighborhood Nº of x^* and a positive constant ν, such that for all x_k

S

Nº, all ρ and all σ_k for which

it follows that Φ(x_k, σ_k ) = 0 and QP(x_k, B_k, σ_k, ρ) has a feasible solution d at which the predicted reduction satisfies

Proof. By Lemma 3.11, there exists a neighborhood N¹ of x^* and positive constants ν, such that for all x_k

S

Nº and all ρ and σ_k for which ν V (x_k) < ρ < , it follows that the QP subproblem (1.1) has a feasible solution d at which the predicted reduction satisfies (3.32). Since the global optimality of d ensures that Δq(d) decreases monotonically as ρ decreases, the predicted reduction satisfies (3.32) whenever ν V (x_k) < ρ.

From the earlier proof, if ν V (x_k) < σ_k < , we also have that the QP subproblem (1.1) is consistent by taking ρ = σ_k . It means that the problem (2.4) has the optimal value 0. If σ_k increases to a value larger than , then the feasible region of the problem (2.4) with ρ = σ_k is also enlarged correspondingly and the optimal value is still 0. Therefore, Φ(x_k, σ_k ) = 0 whenever ν V (x_k) < σ_k.

The above two conclusions complete the proof.

As a last preparation for the proof of Theorem 3.14, we proceed as in [8] and show that the loops 2-(4.1)-2 and 2-(4.3)-2 terminate finitely.

Lemma 3.13. Suppose Assumptions (A1)-(A3) hold, then the loops 2-(4.1)-2 and 2-(4.3)-2 terminate finitely.

Proof. If x_kis a KKT point for the problem (NLP), then d = 0 is the solution of QP(x_k, B_k, σ_k, ρ), and Algorithm 2.1 terminates, so do the loops 2-(4.1)-2 and 2-(4.3)-2. If V (x_k) > 0 and Φ(x_k, σ_k ) - V (x_k) = 0, then stop by Step 3 of Algorithm 2.1. In fact, it follows from [2, Lemma 2.1] that V (x_k) > 0 and 0

V (x_k), which means that x_kis a stationary point of V (x) and not feasible for the problem (NLP). If both above situations do not occur, and the loops 2-(4.1)-2 and 2-(4.3)-2 do not terminate finitely, then ρ 0 from Algorithm 2.1. There are two cases to be considered.

Case (i). V (x_k) > 0 and 0

V (x_k).

(a) If i

and c_i (x_k) > 0, then for all d (d

< ρ),

if either

c_i(x_k) = 0 or ρ < . Thus, for sufficiently small ρ, the equality constraints cannot be satisfied and (1.1) is inconsistent. Therefore, Φ(x_k, σ_k) > 0.

(b) If i

and c_i (x_k) > 0, a similar conclusion is obtained.

(c) If i

and c_i (x_k) < 0, a similar conclusion is also obtained.

So, for some ρ sufficiently small, Φ(x_k, σ_k ) > 0 and 0

V (x_k). By Step 4, the procedure executes at 4.2 of Step 4. It follows from Lemma 3.4 that (2.13) holds. By the mechanism of Algorithm 2.1, the loop 2-(4.1)-2 terminates finitely.

Case (ii). V (x_k) = 0. Then Φ(x_k, σ_k ) = 0.

For the inactive constraints at x_k, by a similar argument, it will still be inactive for sufficiently small ρ. Thus, we only need to consider the active constraints.

Since x_kis not a KKT point, there exists a vector s, s = 1, and a scalar > 0 such that ƒ(x_k)^Ts < -, c_i(x_k)^Ts = 0, i

and c_i(x_k)^Ts < 0, i

(x_k) = {i : c_i (x_k) = 0, i

}. We note that q(0) - q(ρs) = -ρƒ(x_k)^Ts - ½ ρ²s^T B_ks. Using bound on B_k, we have q(0) - q(ρs) > ρ - ½ ρ²nM. If ρ < , then q(0) - q(ρs) > ½ ρ. Since ρs is feasible for QP(x_k, B_k, σ_k, ρ), it follows that the predicted reduction (2.9) satisfies

where d is the solution of QP(x_k, B_k, σ_k, ρ). If ρ < , it follows with (3.20) that Δƒ(d) > ηΔq(d) > 0. So the sufficient reduction condition (2.12) for an ƒ-type iteration is satisfied. Moreover, by (3.21), we have

if ρ < η/(γ₂mnM). Thus, x_k+ d is acceptable to x_k. Of course, the upper bound condition (2.8) is satisfied.

Finally, it follows with Lemma 3.9 that for a sufficiently small ρ, an ƒ-type iteration is generated and the loop 2-(4.3)-2 terminates finitely.

Lemma 3.13 together with Lemma 3.4 implies that Algorithm 2.1 is well-defined. We are now able to adapt [8, Theorem 7] to Algorithm 2.1.

Theorem 3.14. Let Assumptions (A1)-(A3) hold. {x_k} is a sequence generated by Algorithm 2.1, then there is an accumulation point that is a KKT point for the problem (NLP), or a non-MFCQ point for the problem (NLP) or a stationary point of V (x) that is infeasible for the problem (NLP).

Proof. Since the loops 2-(4.1)-2 and 2-(4.3)-2 are finite, we only need to consider that the loop 2-6-2 is infinite. All iterates lie in S, which is bounded, so it follows that the iteration sequence has at least one accumulation point.

Case (i). Step 4.2 of Algorithm 2.1 is invoked finitely many times. Then Step 4.2 of Algorithm 2.1 is not invoked for all sufficiently large k.

Sub-case (i). There are infinite V-type steps in the main iteration sequence. Then, from Lemma 3.7, V (x_k) 0 and τ_k 0 on this subsequence. Moreover, there exists a subsequence indexed by k

S_φ of V-type iterations for which x_k

x^*, V (x_k) 0 and τ_{k +1} < τ_k . One consequence is that x^* is a feasible point for the problem (NLP). If MFCQ is not satisfied at x^*, then x^* is a non-MFCQ point for the problem (NLP). We therefore assume that MFCQ is satisfied at x^* and consider the assumption that x^* is not a KKT point (to be contradicted).

From Lemma 3.12, we know that Φ(x_k, σ_k ) = 0 and the step d is an ƒ-type step for all sufficiently large k if ρ > γ V (x_k). So, x_k+d is acceptable to the filter for sufficiently large k from Lemma 3.9 if ρ² < . Thus, by Lemma 3.11 we deduce that if ρ satisfies

then k is an ƒ-type iterate for sufficiently large k.

Now we need to show that a value of ρ > νV (x_k) can be found in the loop 2-(4.3)-2 such that k is an ƒ-type iterate for sufficiently large k. Since τ_k 0 when k(S_φ) +, the range (3.34) becomes

From the definition of S_φ, we know that τ_{k +1} = V (x_k) < τ_k . Because of the square root, the upper bound in (3.35) can be greater than twice the lower bound. From Algorithm 2.1, a value ρ > ρ_min is chosen at the beginning of each iteration, then it will be greater than the upper bound in (3.35) for sufficiently large k. We can see that successively halving ρ in the loops 2-(4.1)-2 and 2-(4.3)-2 will eventually locate in the range of (3.35), or the right of this interval.Lemma 3.12 implies that it is not possible for any value of ρ > νV (x_k) to produce an V-type step. Thus if k(S_φ) is sufficiently large, an ƒ-type iteration is generated, which contradicts the definition of S_φ. So x^* is a KKT point.

Sub-case (ii). There are finite V-type steps in the main iteration sequence. Then, there exists a positive integer K , such that for all k > K , x_kis an ƒ-type iteration. So {ƒ(x_k)}_k>K is strictly monotonically decreasing. It follows from Lemma 3.6 that V (x_k) 0 and hence, any accumulation point x^* of the main iteration sequence is a feasible point. Since ƒ(x) is bounded, we get that

It follows that Δƒ(d_k) 0, k > K as k +.

Now we assume that MFCQ is satisfied at x^* and x^* is not a KKT point. Similarly to sub-case (i), when

ƒ-type iterations are generated. The right-hand side of (3.37) is a constant, independent of k. Since the upper bound of (3.37) is a constant and the lower bound converges to zero, the upper bound must be more than twice the lower bound. So a value of ρ will be located in this interval, or a value to the right of this interval. Hence, ρ > min{½, ρ_min}. Since the global optimality of d ensures that Δq(d) decreases monotonically as ρ decreases, by Lemma 3.11, Δq(d) > min{½ , ρ_min} holds even if ρ is greater than . This together with (2.12) yields that ƒ(d_k) > η min{½,ρ_min}, which is a contradiction. Thus, x^* is a KKT point.

Case (ii). Step 4.2 of Algorithm 2.1 is invoked infinitely many times. Then it follows from Lemma 3.5 that there exists an accumulation point x^* of {x_k} such that 0

V (x^*), where is an infinite index set satisfying that any k

is a V-type iteration number and _{k +1} = V (x_k+ t_kd_k). If V (x^*) > 0, then x^* is a stationary point of V (x) that is infeasible for the problem (NLP). If V (x^*) = 0 and MFCQ fails to be satisfied at x^*, then x^* is a non-MFCQ point for the problem (NLP). If V (x^*) = 0 and MFCQ holds at x^*, then we prove that x^* is a KKT point for the problem (NLP).

Suppose x^* is not a KKT point for the problem (NLP), but V (x^*) = 0, and MFCQ holds at x^*. Then, from Lemma 3.5, _k 0 on this sequence. Therefore, for all k

, V (x_k) < _k 0 and _{k +1} = V (x_k+ t_kd_k). By Step 4.4, at any new iteration the first trial radius ρ is greater than or equal to ρ_min. So σ_k > γρ_min is true from the fact σ_k (γρ, ρ). Combining this with the fact V (x_k) 0 as k() yields νV (x_k) < σ_k < ρ for all sufficiently large k. It follows with Lemma 3.12 that Φ(x_k, σ_k ) = 0 and (3.32) is satisfied for all sufficiently large k

. Similar to Sub-case (i) of Case (i), k is an ƒ-type iterate for sufficiently large k

. Hence Step 4.2 could not be invoked at this iteration and _{k +1} = _k which contradicts the definition of . So x^* is a KKT point.

4 Numerical results

We give some numerical results of Algorithm 2.1 coded in Matlab for the constrained optimization problems. The details about the implementation are described as follows:

(a) Termination criteria. Algorithm 2.1 stops if

(b) Update B_k. Initiate B₀ = I , where I is the identity matrix with appropriate dimension. Update B_kby the BFGS formula with Powell's modifications [19], which is described as following:

set

where

and

(c) The parameters are chosen as: ρ₀ = 5.0, ρ_min = 10^-4, η = 0.1, γ₁ = γ₂ = 2 × 10^-4, ε = 10^-6 , ₀ = 10 max{1, V (x₀)}, σ_k = 0.9ρ for all k = 0, 1, 2 ... and all ρ > 0.

Firstly, the following examples is from [29]. Since EXAMPLE 3 in [29] is unbounded below, we do not give it here. The numerical results of other examples are described in the following.

EXAMPLE 1

x^* = 0, ƒ (x^* ) = 0. The iteration number of Algorithm 2.1 is 2.

EXAMPLE 2

x^*= (1.224745, 1.224745, 1.224745, 1.224745)^T , ƒ(x^*) = 6. The iteration number of Algorithm 2.1 is 6.

EXAMPLE 4

x^*= (0, 0, 2)^T, ƒ (x^*) = -2. The iteration number of Algorithm 2.1 is 4.

Compared with the results in [29, 28], the computation at each iteration in this paper is less than those in them.

Except for above numerical experiments, we also test some examples from [12]. We compare these numerical results with those in [13]. The detailed results of the numerical tests on these problems are summarized in Table 1, where NIT, NF, and NG represent the numbers of iterations, function, and gradient calculations, respectively. The problems are numbered in the same way as in Hock and Schittkowski [12]. For example, HS022 represents problem 22 in Hock and Schittkowski [12].

For Problem HS022, Φ(x₀) = 0.3941 implies that QP problem is inconsistent at the first iteration. After this, all QP problems are consistent in the subsequentiterations. Similarly, for Problem HS063, only Φ(x₀) = 1.25 implies the same situation as Problem HS022. Except for Problem HS022 and HS063, for all the other problems in Table 1, their QP problems are consistent in all iterations.

The above analysis shows that our algorithm deals with inconsistent QP problem effectively and is comparable to the algorithm in [13]. So, the numerical tests confirm the robustness of our algorithm.

Acknowledgement. We are deeply indebted to the editor Prof. José Mario Martínez and two anonymous referees whose insightful comments helped us a lot to improve the quality of the paper.

Received: 10/III/08.

Accepted: 03/IV/09.

#753/08.

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*

This research is supported by National Science Foundation of China (No. 10771162), Talents Introduction Foundation (No. F08027) and Innovation Program of Shanghai Municipal Education Commission (No. 09YZ408).

Publication Dates

• Publication in this collection
  08 July 2009
• Date of issue
  2009

History

• Accepted
  03 Apr 2009
• Received
  10 Mar 2008