optimality tests for partitioning and sectional search algorithms

Optimality Tests for Partitioning and Sectional Search Algorithms

Vira Chankong

Department of Systems Engineering

Case Western Reserve University

Cleveland, Ohio 44106

Transmitted by L. Duckstein

ABSTRACT

This paper revives a familiar idea for use in solving optimization problems, particularly unstructured nonlinear programs with many variables. The idea involves partitioning variables into groups and successively performing a sectional search with respect to one group at a time. The paper addresses various difficulties normally associated with this type of procedure, and proposes ways to solve them. These include testing optimality of the point at termination, and developing ways to restart the process upon finding that the terminating point is not optimal. The paper also briefly discusses some useful implications of the results on how to group variables in unstructured nonlinear programs so as to improve convergence.

1. INTRODUCTION

Partitioning is a well-known strategy used in numerical optimization and other numerical methods. It involves dividing variables into groups and dealing with each group separately while providing a mechanism for integrat- ing individual results to achieve the desired solution of the original problem. Problems that may be too large and complex to solve directly may thus be handled through a series of smaller and often less complex subproblems. In principle, therefore, partitioning attempts to induce two forms of simplification: size reduction and structural simplification.

While size reduction is always achieved, the degree of structural simplification achievable in each subproblem depends on how much structure the original problem has. For large linear or “semilinear” optimization problems, the Ritter [31], Rosen [32, 331, Benders [3], and generalized Benders [12, 351

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partitioning methods and the Danzig-Wolfe decomposition method [8] fully exploit both forms of simplification. For example, in solving mixed-variable problems in which each function in the objective function and constraints consists of the sum of linear terms and a function of integer variables, the Benders partitioning method [3] partitions the variables into two groups: “linear” variables and integer variables. One of the resulting two subproblems is therefore a linear program, while the other is an integer program. Each subproblem is solved in a specialized fashion to take advantage of its special structure.

This paper reexamines the most primitive form of solution strategy that is based on partitioning. After dividing the variables into groups, the problem is solved sequentially and cyclically with respect to one group at a time, while holding the other groups at their current values. This strategy, termed cyclic sectional search throughout this paper, has been used with some success in unconstrained nonlinear programs as well as large-scale constrained mathematical programs. Typical examples in the unconstrained case include the cyclic coordinate (univariate) search and its variants (see [2O]), and the Gauss-Southwell method [lo]. Examples in the constrained case include several successive-approximation algorithms for solving multistate dynamic- programming problems [19, 21, 27, 34, 391, and a successive-projection method by Han [16].

A variation of the cyclic sectional search can be found in, for example, [l] and [5]. In this approach, when solving a given subproblem, the optimal solution is expressed as a function of variables in the succeeding subproblems. This functional form of solution is then substituted in the next subproblem and the process repeated. When applicable, the approach helps reduce the number of cycles of sectional search and ensures optimality upon termination [6]. However, it only works for special problems where it is feasible and practical to express optimal solutions of subproblems as func- tions of their parameters.

The cyclic sectional search aims primarily at size reduction and does not assume any problem structure (although the existence of some structure may be useful in deciding how variables should be grouped and how each subproblem should be solved more efficiently). We are therefore interested in it as a potential solution strategy for solving large unstructured mathematical programs. Recent attempts in the development of large-scale optimization methods for general mathematical programs have been based on simplification devices other than partitioning [ll, 221. These devices include: approximation (approximating the original problem by a series of specialized problems), dualization (using duality to temporarily remove complicating constraints and to induce separability of subfiroblems), relaxation (relaxing some constraints), restriction (forcing some inequality constraints to be

Optimality Tests for Search Algorithms 159

active), compactification (efficient handling of original and intermediate data), and mechanized prizing (efficient generation of data needed in each iteration). Many solution algorithms have been developed based on these devices. Successive linear programming [2, 14, 291, successive quadratic programming [ 7, 301, and separable programming [22] use the approximation strategy; various forms of decomposition-coordination techniques [18, 22, 26, 28, 36-38, 411, multiplier methods [4, 17, 401, and augmented Lagrangian methods [9, 24, 251 use the dualization concept; and the generalized reduced-gradient method [23, 251 is based on the idea of restriction and relaxation. Reference [23] discusses strengths and weaknesses of these methods.

In this paper, we view the cyclic sectional search as an alternative tool for solving large unstructured mathematical programs. To make this possible, a number of issues must be addressed. When applied to constrained problems, the cyclic sectional search terminates at a fixed point which is often nonoptimal. This is a major stumbling block that renders the approach ineffective in a general setting. To address this troubling issue, we discuss in the following sections how to test whether a given termination point is optimal, and what remedial actions to take upon finding that the termination point is indeed not optimal. Since existing algorithms cited in the third paragraph of this section can be viewed as special cases of the cyclic sectional search discussed in this paper, optimality results developed here will also be applicable to those algorithms. Other issues such as how to group the variables (which will effect the convergence properties), how to solve each subproblem, and how to do bookkeeping effectively will not be addressed in this paper, as they deserve a separate in-depth exposition.

The next section sets up in more precise terms the problem to be considered and the notation to be used. Section 3 establishes optimality for commonly used sectional search algorithms and discusses practical steps for performing optimality tests. Section 4 extends the results to more general partitioning schemes and discusses how they may be used to guide the partitioning task. Concluding remarks are given in Section 5.

2. PROBLEM STATEMENT

Consider an optimization problem

P:minf(x)

subject to gj(x) ~0, j=I,...,m,

where x is an n-vector, and f and each gj are assumed continuously

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differentiable. An unconstrained problem is obviously a special case of P (i.e. m = 0). Also, results developed for P can be easily extended to include problems containing some equality constraints.

Suppose the number of variables, n, is large relative to the number of constraints, m, and we wish to solve P by cyclic sectional search by solving a series of smaller problems, each involving a variable of smaller and manage- able dimension. Let xt = (ri ,..., r: ,..., xi> where lci E R”i and n, + . * * + nq = n. The ith subproblem to be solved at the kth cycle is

subjectto gj(ri:xf ,..., x,k_i,x:iii ,..., xi-‘)$O, j=l,..., m.

Let a solution to P/ be x:. This is, in turn, used to temporarily fix the value of xi in the (i + I)st up to 9 th subproblems to complete the kth cycle. The process is stopped if the termination criterion

%k ,Xk-l or ]~rk-rk-‘]l <E, where l >O, (1)

is met, in which case no further improvement can be made with the above sectional search procedure. Otherwise the process continues with the 1st subproblem of the (k + 1)st cycle. The familiar univariate coordinate search is the simplest form of this partitioning strategy. It is a one-variable-at-a-time (ni = 1) search. More general forms include N-variables-at-a-time (each ni = N > 1) and n,-variables-~-a-time (not all n, are equal). The process just described is a standard cyclic sectional search procedure wherein each subproblem has a distinct set of variables. A more general scheme would allow some variables to be shared among some or all subproblems. Typical examples are various successive approximation methods for solving multistate dynamic-programming problems. These include Larson and Korsak [21], Chankong and Meredith [6], Howson and Sancho [19], Rood et al. [34], and Nopmongcol and Askew [27]. In the sequel, both standard and generalized sectional search schemes will be considered.

In this paper, we are interested in the following questions:

Does the condition (1) guarantee that the fixed point x* is an optimal solution to P?

If x* is not optimal, what can be done to improve the solution process?

We will first set up the necessary notation. For convenience, the standard vector ordering convention will be used. For any two vectors y and z of any

Optimulity Tests fw Search Algorithms 161

dimension, y = z, y bz, or y > z if and only if, respectively, yi = zi, yi > zi, or yi > zi for each component i of the two vectors, and y > z if and only if y,z but yzz.

Let Q=(l,..., 4). For each i E Q, let the n X 1 basic vectors of R”; (of which xi is a member) be written as columns of an n X ni matrix Ei, and let Ji = EjE;.

Clearly, Jix represents a projection of any vector x in R” into the subspace R”i. Moreover, if E, consists only of unit vectors along coordinates of R”, then Ji is a diagonal matrix with exactly ni l’s in appropriate positions along the diagonal and with the rest being zeros. Without loss of generality, we assume that Ei and Ji have the above forms and that the variables are grouped, with rearrangement if necessary, so that gi = Jix = (Or,. . . , OT, XT I 7 OT , . . . ,OT) and x = Xi E oJix. Note also the following properties of Ji:

C ]jzz> ljlj=lj~ JiJj=O foreach i+j, C=.L (2) ieQ

We will also use V&r) and Vigj(z) to denote the 1 X ni gradient vectors of f(x) and gj(x> with respect to xi respectively. Vf<r>Ji and Vg&x)J, will thus be 1 X n vectors with at most ni nonzero components made up of V&) and Vigj(x) respectively. Consequently

vf(x) = C v_f(x)li and Vgj(r)= C Vgj(x)Ji. (3) iSQ icQ

Let x* be the fixed point x k satisfying (1). This is equivalent to saying that each XT is a local optimal solution of PT. Then, by the Fritz-John theorem, there exists, for each i E Q, A: E R and Ai E R” such that

hqViTf(X*) +V,Tg(x*)A,=O

or

AqjiVTf(x*) + JiVTg( x*)Ai = 0, gT(x*)A, = 0, (A;,A;) B oT, (4)

where g(x*) is an m-vector (g,(x*),..., g,(r*))T and V,g(x*) and Vg(x*> are, respectively, m X n, and m X n Jacobian matrices with each row being the corresponding gradient of gj(x*), j = 1,. . . , m.

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3. OF’TIMALITY TESTS FOR STANDARD CYCLIC SECTIONAL SEARCH

First we discuss optimality conditions for x*. An obvious sufficient condition for 1c* to be a stationary point of P is that a projection of any feasible direction from x* in R” into the subspace R “i is always a feasible direction of Pi*. More precisely, let D c R” and Di C R”: represent the sets of feasible directions emanating from x* of P and Pi* respectively. Then x * is a stationary point of P if, for any d E D, there exists dj E Di, i = 1,. . . , q, such that d = Ci E odj or d = Jid for each i E Q, where 2: = (Or ,..., OT,d; ,..., OT>. A proof of this result follows an observation that x* is a point that meets the termination criterion (1). Hence each x7 solves (at least locally) the corresponding PF. Consequently, for each i E Q,

V,f(x*)d,>O for djE Di or Vf(r*)Jid>O forall d E D.

Hence, Ci E oVf(x*)J,d = Vf(r *)d > 0 f or all d E D. Clearly, if the inequality is strict, optimality of x * for problem P is guaranteed. Otherwise, a further condition such as convexity of f must be imposed to ensure optimality.

Three obvious cases within which the above sufficient condition is always satisfied are

(i) x* is interior to the constraint set X of P, (ii) X equals the Cartesian product of Xi,. . , , X,, where Xi is the con-

straint set of P,I”, and (iii) X n iV(x*) = rTi E IXi n A%*), Z c Q, where rTTi E r represents the

Cartesian product of all sets indexed by I.

Case (ii) requires that the feasible region D of P be decomposible into 9 independent subregions Di, (e.g, all constraints are bounds of individual variables with no coupling constraint involved). Case (iii) imposes similar requirements to case (ii), but only within a neighborhood of x*, and it may involve only some Di.

Similar results have been proved for some specific classes of the cyclic sectional search procedure. See, for example, [I91 and [21] for results concerning multistate dynamic-programming algorithms.

The sufficient condition stated above is a condition on the constraint set and does not involve the value of the gradient of f at x*. It is thus useful only for a restricted class of problems. The following result establishes necessary and sufficient conditions for x* to be a stationary point of P in a more general setting.

0ptimulity Tests for Search Algorithms 163

LEMMA 1. Assume that x* satisfies (0, and hence (4). Zf x* is a local

optimal solution to P, then there exists (A:, AT) satisfying (4) such that

(a) either A: = 0 f or each i E Q or A: > 0 fw each i E Q, and

(b) either A, /A: = A, /At = . . * = Aq/A:, if each A: > 0, or PiA, = . . . =P,A,#O forsome@,> ,..., p,>O,ifeach Aq=O.

Conversely, conditions (a) and (b) together imply that x* satisjies the

Fritz-John (necessary) condition fw optimality for the problem P.

PROOF. If x* is locally optimal for P, then the Fritz-John conditions are satisfied. That is, there exists (A’, Ar) > Or such that

A’Vrf(r*)+Vrg(r*)A=O and gr(x*)A=O. (5)

Premultiplying the first part of (5) by Ji yields

A”Jiv~f(r)+JiVTg(r)A=O and gT(r)A=O. (6)

By putting A: = a,A” and Ai = aiA for some LY~ > 0, we can see that (4) is satisfied by (Ay,AT) f or each i. With this choice, condition (a) follows, whether A0 = 0 or A0 > 0. Condition (b) also easily follows: if A0 > 0, then A: > 0 and Ai /A: = A/A0 for each i; on the other hand, if A0 = 0, then for each i E Q, AT = 0 and A = Ai /oi = Pihi = 0, where pi = l/a, > 0. We also note that A > 0 for each i, since Ai > 0. [Note that each Ai cannot be identically zero, since if it were, (A:, AT) = Or, which violates (4).] This proves the necessity part.

To prove the sufficiency condition for stationarity of x*, we first assume that (A:, AT) satisfying (4) exists. If A: > 0 and Ai /A: = A for each i, (4) becomes

JiVTf(x*) + JiVTg(x*)A = 0, A>O, and gT(x*)A=O. (7)

Summing (7) over all i = I,. . , q, and noting that Cq= IJi = I, we have

VTf(r*)+VTg(r)A=O, A>O, and gT(x*)A=O, (8)

which is the Fritz-John condition for P with A0 = 1. If, on the other hand,

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A: = 0 and Ai /cxi = A # 0, cri > 0 for each i, then (4) becomes

A”JiVTf( x) + JiVTg( r)A = 0, A0 = 0, A>O, gr(x)A = 0. (9)

Again summing the first part of (9) over all i yields

A”VTf(r)+VTg(x)A=O, A”=O, A ~0, (IO)

which is again the Fritz-John necessary condition for optimality of P at x*. This completes the proof.

The necessity part of the above result implies that if there is at least one subproblem, say i, with A: > 0 for all (A:, AT) that satisfy (4), and if there is at least one other subproblem, say j, such that all (A:, AT) that satisfy (4) have A; = 0, then x* cannot be optimal.

We also emphasize that the sufficiency part of the above result is only for stationarity, not optimality. Optimality of x* will be guaranteed by (4), (a), and (b) only if further conditions such as some form of convexity on f or second-order sufficiency conditions are added.

If we further assume that x* is also a regular point of the constraint of P, A0 in the Fritz-John conditions (5) must be strictly positive, and (5) can consequently be converted to the well-known Kuhn-Tucker conditions by dividing through by A’. This observation can be easily extended to modify Lemma 1 to account for the regularity assumption on X* as follows.

THEOREM 1. Assume that x* satisfies (11, and hence (4), and that x* is a regular point of the constraint of P. Then x* is a stationary point of P if and only if there exists A 2 0 such that the Kuhn-Tucker conditions

J,Vrf(x*)+J,V’g(x*)A=O and gT(x*)A=O (11)

are satisfied at x* fw each subproblem i.

The proof is trivial from Lemma 1 and will be left to the reader. We note that the sufficiency part does not really require the regularity assumption.

Since optimality implies stationarity, the necessary condition in Theorem 1 is also a necessary condition for optimality of x*. If r* is a local minimum of P, (4) always holds. Under the hypotheses of Theorem 1, there must exist, for each i, (A:, AT) > OT such that (i) A: > 0, and (ii) Ai /A: = A, where A is a fixed vector, to ensure stationarity of r* in the problem P. On the other


hand, failure of either (i) or (ii) indicates that x* cannot be a stationary, let alone an optimal, point of P.

To illustrate Theorem 1, consider the following strictly convex program

s.t. g:x,+x,+x,~l,

where a is a real number. After partitioning the variables as (ri, x2) and xs, the two corresponding subproblems at the initial point (x,, x2, x,) = (4, - $, f) are

P,:min(x,-4)2+(~2 -2)”

s.t. g’:x,+x,<l-x,0=;

and

s.t. g:x,<l-+x,0=+.

Note that <x~,x~,x!$ is a regular point of the constraint g of P. Note also that <x:, x:> is also a regular point of g’ of Pi and is clearly the unique global minimizer of Pi, with the corresponding multiplier of g’ being A’ = T.

Now if a = 3, ~3” = $ is the unique global minimizer of P2, with the corresponding multiplier of g being A” = T. Equality of A’ and A” ensures that (x,“, rg, x,“) is the (unique global) minimizer of P, by the sufficiency part of Theorem 1. That this is indeed the case can be verified by solving P directly.

If +j < a < 3, again ~3” = + is the unique global minimizer of I’,. But now the multiplier g” is A” = 2(a - $), which is nonnegative but not equal to A’. In this case (r p, x:, x,“) is not a minimizer of I’, although the cyclic sectional search starting with this point will also terminate at that same point. This example also illustrates how easy it is for the cyclic sectional search to fail even when applied to a simple convex program.

To complete the illustration, let 0 < a < f. In this case the unique global minimizer of P2 is rs = a and the corresponding A” = 0. Thus the cyclic sectional search will not terminate at (x,“, x&x!$ and will continue to the

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next cycle by solving P, with the constraint g’ being replaced by x1 + x2 Q 1 - a. The process terminates after this cycle with the following subproblem solutions:

3-a -1-a PI: x1=2> x2=-, h’=5+a,

2

Pz: xg=a, A”=O.

Again, according to Theorem 1, this is not a minimizer of P. In essence, the above results indicate a close relationship between opti-

mality of a point satisfying (1) and the existence of a corresponding set of multipliers that is common for all subproblems. It is therefore of practical interest to find out when and how such a set of multipliers can be found. Further discussion on this and other issues will benefit from the following notation. Let there by p (nondegenerate) binding constraints of P at x*, and let A be a p X n matrix with each row being the gradient of a binding constraint with respect to x. It is clear that there are also exactly p binding constraints at r’ of each subproblem Pi. Let Ai be a p X tri matrix with each row being the gradient of a binding constraint of P with respect to xi. Then A, Ai, and Ji are related in the following ways:

where Oi is a p X n, zero matrix;

4 A+, 1 -** ) Ai 1 .** ( A,)= CAli;

i=l

AAT=A,A;+ a** + AqA’g = 5 A,AT; i=l

(12)

(13)

(14)

and

AJiAT = AJ,(]TAT) = A,A:. (15)

With this notation, the Fritz-John conditions [i.e., (4) for PT, and (5) for P] and the Kuhn-Tucker conditions [i.e., (11) for PF] can be rewritten more compactly by replacing Vg(x*> with A, and all m-vector multipliers A with

Optima& Tests for Search Algorithms 167

the corresponding p-vectors of multipliers (which, with minimal ambiguity, will also be denoted by A), and by removing the now redundant complemen- tary slackness conditions. In particular, the Fritz-John conditions for Pi* at XT in (4) and for P at x* in (5) become, respectively,

AyJiVrf( X) + JiArAi = 0, @;,A;) > OT, (da)

and

A”Vf( X) + ATA = 0, (A;,A~) a oT. @aI

To see some practical implications of the results stated thus far, consider the following situation: A cyclic sectional search procedure is carried out until the point x* is reached at which (1) is satisfied, and in the process of solving each Pi* for x:, a set of multipliers (A:, AT) satisfying (4a) is also obtained (as the dual variables of the corresponding constraints). Many cases can arise:

1. Either A:>Oand Ai/Ap=A foreach l~Q,or A:=Oand a,A,=A for some oi > 0 and for each i E Q.

2. Either A: = 0 for some or all i E Q, or A: > 0 for all i but Ai /A: # Aj/Ag for some pair i, j.

3. A: = 0 for each i E Q, but for some pair i, j, oiAi # ojAj for any (Yi’ aj > 0.

4. A: = 0 for some but not all i E Q.

Case 1 implies that x* is a stationary point, and hence a candidate for an optimal solution of P by virtue of Lemma 1 and Theorem 1. It is guaranteed optimal if some appropriate condition such as convexity of f is further imposed.

For cases 2 to 4, r* may or may not be stationary point of P, depending on whether we can find another set of multipliers having similar properties to case 1. If X* and ~7 are regular points of their respective subproblems, neither can have any set of multipliers satisfying (4a) other than the one already found. In this case, the required optimality conditions (case 1) can never be achieved. Hence x* cannot be an optimal solution to P.

If the regularity condition of some pertinent subproblems is not met, there remains the possibility that case 1, and hence stationarity of x*, may be satisfied by some alternative set of multipliers. It is this situation that we now discuss.

Several cases can be considered. We focus on one that is simplest to treat and implement and is perhaps most likely to occur in practice. Consider case

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2 with the additional assumption that x* is a regular point of the constraint

set of P. The following result will be useful.

THEOREM 2. For each i E Q, let xr solve Pi*, and let the corresponding

set of multipliers (Aq,AT) f oun d in the process of solving P,* fw XT be such

that the conditions in case 2 prevail. Assume that x* is a regular point of P.

Then there exists, fn- each subproblem, a set of multipliers (A:, AT) with the

properties that A: > 0 and that hi /A: equals the same constant vector fw

each i if and only if the vector A > 0, where

A=(AAr)-1 c AiA$. (16) iEQ E

PROOF. To show the sufficiency part, let A given by (16) be nonnegative. We will now show that A! = 1 and fii = A constitutes a required set of multipliers. Note that <A:, A:) = (1, AT> > Or and that

AATA= c AjA$ = A c JIAT$, .iEQ .I .iGQ I

or

ATA- c JjAT$ jsQ J

Since A is of full rank due to regularity of x*, we have

ATA - c JiAT; = 0. .iEQ 3

Premultiplying by Ji and noting that JiJj = 0 for i # j, we have

JiATA - ,iAT; = 0. 1


Since (A:, AT) must also satisfy the Fritz-John conditions (4a), we have

0 = JiVTf( X*) + JiAT$ = JivTf( ‘I+ JiATA7

I

(17)

implying that ($, A:) = (1, AT) is indeed a required set of multipliers for each subproblem.

The proof of the necessity part is essentially the reverse of the above argument. Let there exist (i:, 17) satisfying the Fritz-John conditions (4a) for each subproblem such that A: > 0 and ffi /AT = A. Clearly, A > 0, and

JiVTf( X) + j,ATA = 0 for iEQ.

Replacing each JiVTf(x) with - JiATA, /A: in view of (17) and summing over all i E Q yields

- c liAT$+ATA=O, iEQ I

or

ATA = c J,A’; iEQ I

Premultiplying by A and noting that AAT is invertible, we have

A=(AAT)-r c A]iAT;=(A4T)-1 c A,A$. iEQ 1 ieQ I

This completes the proof.

One practical implication of the above result is as follows. Suppose the optimality test based on the original sets of multipliers (A!,AT) fails to produce a definite conclusion. Under a regularity assumption on x*, not only does Theorem 2 provide a convenient means of checking whether a common set of multipliers exists, it also furnishes a way of computing such multipliers from the original sets. First A is computed using (16). Then it is checked for nonnegativity. If A is nonnegative, x* is claimed to by a stationary point of P

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by virtue of Theorems 2 and 1. Otherwise, x* is not optimum, and the vector A can be used to make further improvements, as will be discussed later.

To illustrate, we modify the example given earlier to read

P:min(x, -4)2+(x2 -2)2+(~3- a)”

s.t. g,:r,+x,+x,<l,

g,:(x, -2)2+(X,)2 <$.

With the same partitions and initial point as used earlier, the two subproblems are

&?:(x, -2)2+(x2)2 <;

and

s.t. g,:x&.

Let a = 3. Again, the cyclic sectional search starting at x0 = <$, - t, i)’ will immediately terminate at x0, with (xy, xi) = (4, - $1 being the unique global minimizer of P,, and xi = $ that of P2. Furthermore, xi is a regular point of P2, and A’; = F is the unique Kuhn-Tucker multiplier of g”. (Since g, does not appear in I’s, h”, can assume any value.) However, <x:, x,“) is not a regular point of P,; the Kuhn-Tucker multipliers associated with g, and g, are not unique. Solving P, numerically can yield any values of the multipliers as long as they satisfy

Suppose I’, is solved numerically (by commercial NLP software) and A’, = y, A’, = 1 are reported as values of Kuhn-Tucker multipliers of g; and gh respectively. Clearly, since A’, # A>, we have (A’,, A’,) # (A’;, XL) regardless of the choice of A’;. Is x0 a minimizer of P? The answer is yes if we can


construct a set of Kuhn-Tucker multipliers common to both P, and Pz by choosing appropriate A’,, A’,, and A”,. (A’; must be fixed at 9.) This can be done by using (16). At x0, we note that

).

and

1 A,= ,() ( 1

Using

we have

which is nonnegative. Thus, according to Theorem 2, ~a is a minimizer of P,

and A is the required vector of Kuhn-Tucker multipliers. Indeed replacing A, /A! in the above expression by A’= (A’,, A’,jT, we see that

h = ($ - $I’, + A’2 +4)T,

which is nonnegative as long as

- A’, + $A’, + F >, 0,

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Upon examination of the Kuhn-Tucker conditions of P, at (x~,;x~), we observe that the Kuhn-Tucker multipliers (A’,, A’,) must satisfy

which clearly implies nonnegativity of A. Now consider a = 1 (or any value between $ and 3). The solutions of P,

remain unchanged, and xi = i remains the unique global minimizer of Pz. But this time Xr = 4. Carrying out a computation similar to the above case with A, /At = (A’,, A’,), we have

The second component of A is negative (viz. -3) for all A’,, A’, that satisfy the Kuhn-Tucker conditions of P, at <x:, x,“). Hence x0 is not optimal for P.

We note that regularity of x* is a mild assumption and that regularity of rr for any subproblem Pi* will indeed ensure regularity of x*, but not vice versa. Let us now drop the regularity assumption and consider the case where Al may be equal to zero for some i. For notational convenience, let Z = {iii E Q, Ai > O} and Q - I = (iii E Q, A: = 0).

THEOREM 3. For each i E Q, let xr solve P,i”, and let the corresponding

set of multipliers (A:, AT) satisfying (4a) be found in the process of solving Pi*.

(a) There exists, for each sybproblem, a set of multipliers (A:, AT) satisfy-

ing (4) such that A: > 0 and Ai /A: = A > 0, if and only if the system

ATA= xI,AT$- C JiVTf(x*) and A > 0 (18) is1 t iEQ-2

has a solution, where JiVTf(x*) is an n-vector made up of the directly

computed Vf (x *> and zeros. (b) There exists, fm each s&problem, a set of multipliers (0, AT‘, satisfy-

ing (4) such that aiii = A fm some q > 0, if and only if the system

ATA = 0, A>0 (19)

has a (nontrivial) solution.

Optima& Tests fm Search Algorithms 173

PROOF. The proof of part (a> is similar to that of Theorem 2. If (18) has a solution A > 0, the set of multipliers (A:, AT) = (1, AT> will satisfy the Fritz- John conditions for each subproblem, since (1, AT) > Or and

Ji ATA = + JjATAi/A: = - JiVTf(x*) if i E Z,

- JYf(x*) if iEQ--I.

On the other hand, the existence of (A:, AT) satisfying (4a) with R > 0 and ii /q = A > 0 for each i implies, after dividing (4a) by 8, that

JiATA= -JivTf(x*). (20)

By substituting JiVTf(x*) by - JiATAi /A: f or each i E I, directly computing ViTf(x*) for each i E Q - I, and summing (20) over all i E Q, we have (19) as required.

Part (b) can also be easily shown in a similar manner and will be left to the reader.

We observe that Theorem 2 corresponds to part (a) of Theorem 3 when I = Q and x* is a regular point of P. If Z # Q, but x* is still a regular point of P, part (b) of Theorem 3 is not applicable and the optimality test reduces to testing whether A = (AAT)-‘[& E ,JiATAi /A: - Ci, Q_lJiVTf(~*)] is nonnegative. The only difference between this case and Theorem 2 is that it is no longer possible to estimate the segments JjVTf(x*) of VTf(r*) in terms of the multipliers (A:, AT) using (20). The value of JiVTf(x*) will need to be obtained through direct computation of Vi f( x *>.

If x* is not a regular point of P, both tests in parts (a) and (b) of Theorem 3 may be needed, irrespective of whether Z = Q or not. If either (18) or (19) has a solution, then we conclude that x* satisfies the Fritz-John necessary conditions for optimality of P. If neither has a solution, x* is clearly not a local optimal point of P due to Lemma 1.

To check the feasibility of (18) for the case where x* is not a regular point of P, we may apply phase 1 of the simplex method to (18) directly. However, a more efficient procedure is to apply phase 1 of the simplex method to an equivalent system

AATA = AC and cT(ATA - c) = 0, A > 0, (21)

174

where

VIRA CHANKONG

It is easy to show that (18) and (21) have the same solution set. Also, n being much larger than p, the dimension of (18), n X p, is much larger than that of

(20, (p + 1) X p. To check the feasibility of (19), which is needed only for the case where

x* is not a regular point of P, we again apply the first phase of the simplex method to check whether an equivalent but smaller (p X p) system

AATA = 0, A >o, (22)

has a nontrivial solution. Having outlined a procedure for checking optimality of a point x* which

satisfies the termination criterion (1) of a sectional search algorithm, we next address another practical question. Upon discovering that x* is not optimal, how should we proceed to get away from x*? Two strategies may be tried: One is to use the current information to find a feasible descending direction from r*, perform a line search, and resume the sectional search operation at the new point with either the previously used or a new partitioning of variables as appropriate. Another approach, which may be termed adaptive partitioning and sectional search, is to use the current information to formu- late new partitions of variables and resume the sectional search operation with these new partitions. In what follows, we discuss the first approach further. The second approach, which intimately involves the discussion of how best to partition variables so as to achieve a good rate of convergence, will be explored elsewhere.

Consider now the situation where x* satisfies (I) but is found not to be an optimal point of P due to the nonexistence of a common set of multipliers (A:, AT) for all subproblems Pi* [although an individual set (A!, AT) for each subproblem exists]. We now wish to utilize the knowledge of x* and (A:, AT> to find a direction of descent so that a line search can be performed.

A procedure that seems most appropriate, requiring minimum assump- tions and being fairly simple to implement, is a modified Zoutendijk’s feasible direction approach [24] with antijamming feature. With this approach, a feasible descending direction d is found by solving the following

Optimality Tests for Search Algorithms

LP problem:

175

s.t. vf(x*)d-q<o,

Since x* is not an optimal point to P, there must exist a strictly descending direction d such that Vf(x*)d < 0 w ic h h can be used in the next line search.

If the value of Vfcr*) is not already available, some or all of its segments, J,Vrfcx*), may be computed using (A!, AT) and Ai as given in (20) provided that A! # 0. In general,

vT’(x*) = c JiAT$ + c JivTf(X*b (23) i6I I iEQ-I

where, again, JiVT’(x*), i E Q - 1, is computed directly through the computation of Vf(r *).

An alternative to Zoutendijk’s feasible-direction approach, when x* is a regular point of P, is the gradient projection method. The projection matrix P at x* is given by P = I - Ar(AAr)- ‘A. Hence the projection of Vf(x*) on the tangent space to the constraint set of P at x* is

d* = - p*v~j-(~*) = [I- A~(AA~)-~A]vT~(~*),

where Vr’(x*) is given by (22) or (23). If d* z 0 and is usable (which is the case if the binding constraints are

linear), we perform a line search along d*, test optimality at the new point, and resume the sectional search operation if the new point is not optimal. If d* z 0 but is not usable (which is usually true if the binding constraints are nonlinear), we still proceed with a line search along d*, after which the resulting solution is projected back to the constraint surface. If d* = 0, we

compute A using A = (AAT>-‘AVrf(x*), w h ere again VTf(x*) is given in (22) or (23). At least one component of A should be negative; otherwise r* would satisfy the Kuhn-Tucker conditions for P. Following the usual gradient projection method, a binding constraint corresponding to the most negative

176 VIRA CHANKONG

,. component of A is dropped from A to form A, and the new projection matrix P^ = i- ffT(BT)-‘A and the new direction d^= PVTf(x*> are computed.

Regularity of r* will ensure that d^ # 0 and that Vf(x)o! < 0. The direction d^ can thus be used in the next line search.

In comparing the two methods, the feasible-direction method, which involves solving an LP problem of size (m + 2) X (m + 1) but does not require the regularity assumption of x*, should be superior to the gradient projection method if the constraints of P are nonlinear. In such nonlinear problems, the latter usually requires considerable amount of work in an attempt to bring the often infeasible solution resulting from the line-search procedure back to feasibility. On the other hand, for linearly constrained problems, where such an effort is not required, the gradient projection method should prove more efficient, since it finds a feasible descending direction without solving an optimization problem.

4. OPTIMALITY TESTS FOR GENERALIZED CYCLIC SECTIONAL SEARCH

We now consider more general partitioning and sectional search schemes. The simplest and perhaps most useful extension is the scheme wherein all variable partitions share a common set of variables x0 of dimension n,. That is, subproblem i will be optimized with respect to r0 and xi and is therefore of dimension n, + ni. If we define Jo, E,, and V,,f(x) in the same way we defined other Ji, Ei, and Vif(x), then we can write the Fritz-John conditions at a local optimal point r,* of subproblem Pi* as follows:

A:( Jo + Ji)vTf( x*) + (Jo + Ji)VTg( “*)hi = O>

or

A;( Jo + Ji)VTf( x*) + (Jo + Ji)ATAi = 0, i=l,...,q. (24)

Results similar to the standard case (Section 3) can now be easily established.

LEMMA 2. x* satisfying (24), and hence (l), is a stationary point of the problem P if and only if there exists (A!, AT) having the following properties:

(i) either A! = 0, fm each i E Q or A! > 0 fm each i E Q, and (ii) there exists a common vector A E RP such that either Ai /A: = A if

A! > 0 fat- each i E Q, or fm some q > 0, aiAi = A if A: = 0 fm each i E Q.


A proof can be patterned after that of Lemma 1 and will thus be omitted. Imposing a mild regularity assumption simplifies the result.

THEOREM 4. Zf (x,, xi) is a regular point of Poi fw at least one i, or x* is

a regular point of P, then x* is a stationary point of P if and only if there

exists A > 0 such that, fw each i = 1,. . . , q,

(lo + Ji)VTf(x*) + (Jo + Ji)ATh = 0. WI

PROOF. The sufficiency part is trivial-and does not require any regularity assumption. For the necessity part: If x* is a regular point of P, the Kuhn-Tucker conditions must be satisfied and (25) immediately follows. If XT is a regular point of Poi, the proof will be complete if it can be shown that x* is also a regular point of P. Regularity of x7 implies that A(J, + Ji> has full rank p and A(],, + J,)A r is positive definite. Consequently, for any x E R”, xTAATx = xTA(Jo + Ji)ATx + Cj + ixTA]jATx is strictly positive, implying that A also has full rank p. Thus, x* is also a regular point of P. This completes the proof.

If we assume further that xa is a regular point of P,,, the terminating condition (1) becomes sufficient for stationarity, as the following result indicates.

COROLLARY 4.1. Zf x0 is a regular point of PO, then x* satisfying (1) is guaranteed to be a stationary point of P.

To prove this, we first observe that as AJ,, is of full rank p due to regularity of x,,, so is A(J,, + Ji) f or each i and A. Also, by premultiplying (24) by Jo and noting (2), we have

A'&VTf(x*) + JoAT& = 0, (26)

(A;,A;)>~, (A;,A;)zo foreach i=l,...,q. (27)

Clearly A: > 0, for otherwise A! = 0 and (26) would be implied Ai = 0, which contradicts the fact that (A:, A:) # 0. Also we have hi/A7 = -(A,,A;>-‘A,,VTf(r> = A f or each i. The required conclusion follows from Theorem 4.

178 VIM CHANKONG

We see that, by including the coupling variables x0 as part of every subproblem, the terminating criterion (1) may indeed become a sufficient condition for optimality. The regularity assumption required is likely to be satisfied if the number of binding constraints p at r* is expected to be small relative to the number of coupling variables rza. Even without such an assumption, the inclusion of x0 in each subproblem should improve the convergence properties of the sectional search algorithm, although it may be done at the expense of making each subproblem larger by n,.

Again from the practical viewpoint, we would be interested in what to do after proceeding with a sectional search procedure and ending up with a point XT which solves each P,,i, i = 1,. . . , q, and the corresponding multipliers (A:, AT). Such multipliers will obviously satisfy (24) or equivalently, for each i = 1,. . . , q,

A(&Vrf( x*) + JoATAi = 0, (28)

A~J,V~~( x*) + J,ATAi = 0. (29)

No further action needs to be taken (apart from ensuring that a stationary point is indeed local optimal) if one of the following cases is true:

(i) Aa=Oand aiAi=A foreach i=l,...,q andsome cri>O, (ii) it > 0 and hi /A: = A for each i, or (iii) A: = JaA“ has rank p.

As noted earlier, case (iii), which co,rresponds to x0 being a regular point of the constraint g(r)], < 0, implies case (ii).

If none of the above is true, we next determine whether there are other sets of multipliers satisfying either (i) or (ii). Results similar to Theorems 2 and 3 will be useful for this purpose. Again, let the sets Z and Q be as previously defined.

THEOREM 5. Let x* sutisfv (0, so that <r,,x~> satisfies (24) for some (A;, Ay) E Rp+’ for each i. Let, without loss of generality, A: > 0.

(a) There exists, fm each subproblem Poi, (R, AT> E Rp+’ satisfying (24) such that A: > 0 and Ai /A: equals a constant vector common for all i=l , . . . , q if and only if there exists a set of multipliers A E RP with A >, 0

Optimal&~ Tests jbr Search Algorithms

and

179

An=JoAT$+ cJ,AT$+, c J,V’f(x*). (30) 1 iGZ t tEQ-1

(b) There exists, fm each subproblem Poi, & E RP and a scalar pi such that pi& = h’ if and only if there exists a (nontrivial) solution to

ATA = 0, h.0, and A#O. (31)

PROOF. (a): To prove the sufficiency part, let (30) have a feasible nonnegative solution A > 0. Setting A: = 1 and hAi = A, we observe that (A:, AT) > 0 and (8, iT> z 0 and that by premultiplying (30) by J,, and Ii, we have

],,ATA=JoAT$=-J,V’f(x*), (32) 1

Ji ATA = +JiATAi/A:=-JiVTf(x*) if iEZ,

- JivTf(‘*) if iEQ--I, (33)

where the last equalities of (32) and (33) are due to (28) and (29). Combining (32) and (33), and rearranging, we have, for each i = 1,. . . , q,

(J~+Ji)v~f(x*)+(Jo+Ji)A~A=o, (34)

which is exactly the same as (24) with &, = 1 and fii = A. For the necessity part, we start with (28) and (29) for some (A!, iy) 2 0

such that A? > 0 and Xi /A: = i for each i. Clearly A > 0 and

.

l$Tf( x*) = - &AT; = - J,,A'i I

and

(35)

~,v’f(~*) = - JA’; = - J,A'i.

I

180 VIBA CHANKONG

Adding (35) and (36) for each i indicates that ho = 0 and hi = fii /R = fi satisfy (30) (with I = Q) as required.

(b): The proof of th is part is similar to part (a) and is left to the reader.

5. CONCLUDING REMARKS

The results presented in this paper provide a basis for developing a convenient scheme for testing optimality of a terminating point of two types of partitioning and sectional search algorithms. The testing scheme essentially involves verifying the existence of a set of Kuhn-Tucker multipliers common to all subproblems, either by comparing the corresponding multipliers generated for each subproblem or by determining feasibility of a set of linear inequalities. Some ideas for using the current information to reactivate the sectional search process once the terminating point is found to be nonoptimal are also discussed.

The two partitioning schemes considered are one that produces disjoint groups and one that produces groups sharing a common subgroup. The former is simpler and more commonly used, but usually converges slowly, if ever. By letting each subproblem share a common set of variables as suggested in the latter scheme, the overall convergence of the algorithm can be improved substantially. However, this is achieved at the expense of increasing the dimension of each subproblem (by the dimension of the mutually shared subspace). This brings up the question of how to actually do the grouping of variables to provide the best compromise between convergence and computational efficiency. While the partitioning is generally obvious for problems with special structures (such as those dealt with by Rosen [32], Benders [3], and Dantzig and Wolfe [S]), it is not so in unstructured problems. This question, along with a detailed analysis of convergence and computational efficiency, an extension of the results to more general partitioning schemes, and an exploration of the idea of “adaptive” or “evolv- ing” partitioning to help improve convergence, is a subject for further research.

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