It has been shown by Fiacco that convexity or concavity of the optimal value of a parametric nonlinear programming problem can readily be exploited to calculate global parametric upper and lower bounds on the optimal value function. The approach is attractive because it involves manipulation of information normally required to characterize solution optimality. A procedure is briefly described for calculating and improving the bounds as well as its extensions to generalized convex and concave functions. Several areas of applications are also indicated

Fiacco, Anthony V.

Kyparisis, Jerzy

English

NASA Technical Reports Server

COMPUTABLE OPTIMAL VALUE BOUNDS FOR
GENERALIZED CONVEX PROGRAMS
N8 -1885 
Anthony V. Fiacco
Department of Operations Research
The George Washington University
Washington, DC
Jerzy Kyparisis
Department of Decision Sciences
Florida International University
Miami, Florida
SUMMARY
It has been shown by Fiacco that convexity or concavity of the optimal value
of a parametric nonlinear programming problem can readily be exploited to calculate
global parametric upper and lower bounds on the optimal value function. The
approach is attractive because it involves manipulation of information normally
required to characterize solution optimality. We briefly describe a procedure
for calculating and improving the bounds as well as its extensions to generalized
convex and concave functions. Several areas of applications are also indicated.
INTRODUCTION
We are concerned here with parametric nonlinear programming problems of the
form
min f(x,t) s.t. g(x,t) > O, h(x,t) = 0 P(t)
X
where f is a real valued function, g and h are vector valued functions, and t is
a parameter vector. The optimal value function of P(t) is defined by
f*(t) = min { f(x,t): x 6 R(t) }
where R(t) is the feasible set of the problem P(t) given by
R(t) = { x : g(x,t) > O, h(x,t) = O }
In this paper we describe a procedure, originally proposed by Fiacco (refs.l,2),
for calculating piecewise-linear continuous global upper and lower parametric bounds
on the convex (or concave) optimal value f*. We also show how these bounds can be
improved in a systematic manner until a desired accuracy, as measured by the
maximal deviation from the optimal value over the interval of parameter values, is
achieved. Extensions of this approach to generalized convex and concave optimal
value functions are discussed as well and current experience with applications is
described.
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https://ntrs.nasa.gov/search.jsp?R=19870009424 2020-03-20T11:35:23+00:00Z
COMPUTABLE BOUNDS ON CONVEX OPTIMAL VALUE FUNCTIONS
Consider the parametric problem P(t) and assume that its optimal value
function, f*(t), is convex. This will be the case if P(t) is a jointly convex
program, i.e. if f is jointly convex in x and t, the components of g are jointly
concave in x and t, and those of h are jointly linear affine in x and t (ref. 3).
The assumptions on g and h can actually be generalized by requiring only that the
map R is convex (ref. 4).
Suppose now that we have evaluated f*(t) and its slope at two distinct values
t I and t_ of the parameter t, where (for simplicity) t is assumed to be a scalar.
Then, global definitional properties of convex functions immediately provide global
parametric continuous, piecewise-linear bounds via linear supports and linear
interpolation on the graph of f* over the line segment (tl,t2). This is illustrated
in figure i.
Practical implementation of bounds calculations requires only the information
provided by most standard nonlinear programming algorithms. In particular, the
solution of the problem P(t) as well as the associated optimal Lagrange multipliers
must be determined for two distinct parameter values. The Lagrange multipliers will
coincide with derivatives of f* in case f* is differentiable and with subgradients
of f* in case when f* is nondifferentiable and convex. In both cases the
multipliers can be used to compute the lower bounds on f*.
Clearly, if f* is convex on the convex set SCE r, then any supporting hyperplane
of the epigraph at any t G S provides a global lower bound on f* over S. Both
upper and lower bounds calculations obviously apply over any interval (tl,t 2) in S,
provided that f* is convex over (tI. t2) A standard technique for studying f*
over (t I, t2) is to consider t(a) = at I + (l-a)t 2 and view f as a fuction of the
scalar parameter a @ (0,i). This allows for the simultaneous pertubation of all
components ti of t, which are now linear affine funtions of the scalar parameter a.
A byproduct of this practical approach is the observation that if the feasible
point to set map R is convex then x(a) = ax I + (l-a)x 2 G R(t(a)) if x I G R(t I) and
x2 @ R(t2). This leads to the simple calculation of a feasible parametric vector
x(a) of a problem P(t(a)) whenever the,condition is met. Hence we also obtain the
upper bound T (a) = f(x(a), t(a)) on f (t(a)) over (tl,t2). Since the calculation
of x(a) does not depend on f, this does not require f to be convex. If f is,jointly
then f is convex and f (a) is a convex bound on or above f andconvex in (x,t), * " --
below the linear upper bound given in figure i.
The parametric bounds on the optimal value function f* described above were
constrained to one-dimensional perturbations of the parameter vector t. However,
it is a simple matter to extend these bounds to multi-dimensional perturbations of
t.
Suppose, for example, that f* is convex on the convex set SeE r and that we are
interested in bounds on f_ for t in some polyhedron M contained in S which is
determined by its extreme points t I, t2, ..., t_. To obtain these bounds we need
only to determine the values and subgradients of f* at _ points tI ..... t_. T_is
• i)information will be available if we compute optimal solutions x (t and Lagrange
multipliers for _ nonlinear programs P(tl), i=l,...,l, similarly to the case of
one-dimensional perturbations. If, in addition, R is a convex map, then we can
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calculate a feasible parametric vector x(a), for the problem P(t(a)) with t(a) 6 M,
as a convex combination of _ solutions x*(ti), i=l .... ,i as well as a sharper convex
-- f*upper bound f (a) on .
The described approach for calculating parametric upper and lower bounds on
convex f* can be extended to the case of concave optimal value function f*. The
well known sufficient conditions for concavity of f* require that f be concave in t
for t G S and the feasible set R(t) = Nofor all t G S (that is R(t) must be fixed).
This result can be generalized to programswith perturbed feasible sets R(t) by
assuming that the mapR is concave (ref. 4).
If we now assume, similarly to the convex case, that f* is concave over the
interval (tl,t 2) and the values and slopes of f* are knownat two distin_points t 1
and t 2, then a linear interpolation on the graph of f* will provide a lower bound
while a piecewise linear upper bound will be determined by the slopes of f*. Figure
2 illustrates these bounds.
REFINEMENTSOFOPTIMALVALUEBOUNDS
In the previous section we described a procedure for calculating piecewise-
linear optimal value bounds on convex or concave f*(t) over the interval (tl,t 2) of
parameter values. Wealso showedthat a parametric feasible solution vector x(a),
a G (0,i), is an immediate by-product of this approach. This remarkably regular
behavior is exploitable in a number of waysas will be shownnext.
Consider a convex f*(t(a)), where t(a) = at I + (l-a)t 2, and view it as a
function of the scalar parameter a 6 (0,i) with upper and lower bounds on f* as
depicted in figure i. Supposethat we solve the program P(t(a*)) at some
intermediate value a* G (0,i). Then, this additional solution of P(t(a*)) enables
us $_ easily calculate sharper piecewise-linear continuous upper and lower boundson . Thesenew bounds on f* along with previous bounds are illustrated in figure
3.
Moreover, we can calculate a more accurate piecewise-linear continuous feasible
estimate x(a) of the parametric solution vector, which in this case is the linear
interpolation between contiguous optimal solutions of P(t(a)) at three values
a=0,a ,I. The feasible solution x(a) allows, in turn, the computation of a sharper
piece-convex continuous upper bound on f*, given by f (a) = f(x(a),t(a)).
Similar sharper piecewise-linear continuous upper and lower bounds can be
computedfor a concave optimal value function f* by solving an additional program
P(t(a*)) at someintermediate value a* G (0,i). The improved bounds will be
analogous to those depicted in figure 3.
It is clear from figure 3 that by repeatedly solving the program P(t(a)) at
intermediate values of a, the bounds on f* may be quickly and significantly improved.
The value a* of the parameter at which the problem was solved is the value where the
deviation between the current upper boundU and lower bound L, i.e., U(a) - L(a),
is the maximumover the considered interval (0,1). This is an appealing choice,
although other choices might,be dictated by other criteria or user interest; eg. it
might be important to know f (a) accurately only for certain subintervals or certain
choices of a.
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EXTENSIONS OF BOUNDS TO GENERALIZED CONVEX OPTIMAL VALUE FUNCTIONS
The approaches for calculating parametric optimal value bounds described
earlier can be extended in several ways to include much wider classes of parametric
programs. This means that optimal value bounds are much more widely applicable than
is apparent from the results of the previous sections.
The first extension is obtained by considering structured classes of generalized
convex and concave optimal value functions. Suppose that map R is convex and that
f is quasiconvex in (x,t) for t 6 (tl,t2). Then, f* is also quasiconvex (ref. 5)
and therefore a constant upper bound of f* given by max {f*(t I) f*(t2)} is
readily available as well as a sharper quasiconvex upper bound f (a) = f_(a),t(a)).
Additional classes of convex and nonconvex programs for which parametric upper
bounds on f* can be computed are those where the objective function and,
consequently, the optimal value function F* are strongly convex, strictly
quasiconvex and strictly pseudoconvex (ref. 5).
Analogous results can be obtained in the concave case. Assume, for example,
that the feasible set R is arbitrary and fixed and that f is quasiconcave in t for
t G (tl,t2). Then, quasiconcavity of f* follows (ref. 5) and min {f*(tl), f*(t2)}is
a constant lower bound on f*
The second extension is possible by considering generalized convex programs
which are transformable into standard convex programs. Consider program P(t) with a
convex feasible map R and an F-convex objective function f. That means that the
composed function fF (x,t) = F{f(x,t)} is convex in (x,t) where F is a continuous,
one-to-one function (ref. 6). Thus the optimal value function fF* of a modified
problem PF(t)
min fF(x,t) s.t. g(x,t) _> O, h(x,t) = 0 PF(t)
is convex and therefore piecewise-linear upper and lower bounds on fF , given by
L(t) j fF*(t) < U(t) can be calculated. Then, since fF*(t) = F{f*(t)}, one
immediately obt--ains the following bounds on f*(t) (provided that F is nondecreasing):
F -I {L(t)} < f*(t) < F-I {U(t)}. These bounds are in general nonlinear and
nonconvex but, nevertheless, can be calculated without difficulty once the program
PF(t) has been solved.
EXPERIENCE WITH APPLICATIONS
Several preliminary studies have been conducted to investigate some of the
more immediate computational and practical implications of the outlined approach for
generating global parametric upper and lower optimal value bounds. The procedure
for calculating optimal value bounds for both convex and concave optimal value
functions was implemented by Fiacco and Ghaemi (ref. 7) as an additional module in
the penalty-function based sensitivity-analysis computer program SENSUMT.
Fiacco and Ghaemi (ref. 8) studied a geometric programming model of a stream
water pollution abatement system and calculated bounds on the convex optimal value
(defined as the annual cost of operation) of an equivalent convex program. The
indicated water pollution bounds calculation involved the perturbation of a single
right-hand-side parameter, the allowable oxygen deficit level in the final reach of
the stream, that proved to be the most influential parameter in the prior sensitivity
study.
14
Subsequently, Fiacco and Kyparisis (refs. 9, 10) utilized SENSUMT to calculate
bounds on the optimal value function for the same water pollution abatement model
when 30 (not all right-hand-side) most influential constraint parameters were
perturbed simultaneously. In this application, the optimal value function was not
convex in full neighborhood of the base value of the parameter vector. However,
it was possible to show that the restriction of f*(t) to the subset of parameters
involved in the desired perturbation is convex.
In another study involving the convex equivalent of a geometric programming
model of a power system energy model, to find the turbine exhaust annulus and
condenser system design that minimizes total annual fixed plus operating cost, Fiacco
and Ghaemi (ref. 11) used SENSUMT to obtain bounds on the optimal value function for
a variety of single objective function and constraint parameter changes. A novelty
of this anlysis is the exploitation of problem structure to calculate a nonlinear
lower bound on the optimal value function. In addition, parametric bounds are
computed on the optimal value which is concave for certain perturbations of
objective function parameters.
REFERENCES
io Fiacco, A.V.: Computable Optimal Value Bounds and Solution Vector Estimates
for General Parametric NLP Programs. George Washington University (CWU)
Technical Paper T-451, 1981.
o
.
Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear
Programming. Academic Press, New York, 1983.
Mangasarian, O.L.; and Rosen, J.B.: Inequalities for Stochastic Nonlinear
Programming Problems. Operations Res., vol. 12, no. i, 1964, pp. 143-154.
Q Fiacco, A.V.; and Kyparisis, J.: Convexity and Concavity Properties of the
Optimal Value Function in Parametric Nonlinear Programming. J. Optimiz.
Theory Applic., vol. 48, 1986, pp. 95-126.
1
.
Fiacco, A.V.; and Kyparisis, J.: Generalized Convexity and Concavity Properties
of the Optimal Value Function in Parametric Nonlinear Programming. George
Washington University (GWU) Technical Paper T-472, 1983.
Avriel, M.: Nonlinear Programming: Analysis and Methods. Prentice-Hall,
Englewood Cliffs, NJ, 1976.
o Fiacco, A.V.; and Ghaemi, A.: A User's Manual for SENSUMT: A Penalty Function
Computer Program for Solution, Sensitivity Analysis and Optimal Value
Bound Calculation in Parametric Nonlinear Programs. GWU Technical Paper
T-434, 1980.
So Fiacco, A.V.; and Ghaemi, A.: Sensitivity Analysis of a Nonlinear Water
Pollution Control Model Using an Upper Hudson River Data Base. Operations
Res., vol. 30, no. i, pp. 1-28.
1 Fiacco, A.V.; and Kyparisis, J.: Computable Parametric Bounds for Simultaneous
Large Perturbations of Thirty Parameters in a Water Pollution Abatement
GP Model, Part I: Optimal Value Bounds. GWU Technical Paper T-453, 1981.
15
i0.
ii.
Fiacco, A.V.; and Kyparisis, J.: ComputableParametric Bounds for
Simultaneous Large Perturbations of Thirty Parameters in a Water
Pollution Abatement GPModel, Part II: Refinement of Optimal Value
Boundsand Parametric Solution Bounds, GWUTechnical Paper T-461, 1981.
Fiacco, A.V.; and Ghaemi, A.: Sensitivity and Parametric BoundAnalysis of
Optimal SteamTurbine Exhaust Annulus and Condenser Sizes. GWUTechnical
Paper T-437, 1981.
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/unds on Convex ft2. t
I
/
/
/
F
/
I
Optimal Value bound8 t2
on cOncave f_
fi t
J l
( j
( I
J t I
0 a* i
Figure 3. Improved optimal value bounds on convex f*.
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Computable optimal value bounds for generalized convex programs

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