paretosearch
Algorithm
paretosearch
Algorithm Overview
The paretosearch
algorithm uses pattern search on a set of
points to search iteratively for nondominated points. See Multiobjective Terminology. The pattern search
satisfies all bounds and linear constraints at each iteration.
Theoretically, the algorithm converges to points near the true Pareto front. For a discussion and proof of convergence, see Custòdio et al. [1], whose proof applies to problems with Lipschitz continuous objectives and constraints.
Definitions for paretosearch
Algorithm
paretosearch
uses a number of intermediate quantities and
tolerances in its algorithm.
Quantity  Definition 

Rank  The rank of a point has an iterative definition.

Volume  Hypervolume of the set of points p in objective function space that satisfy the inequality, for every index j, f_{i}(j) < p_{i} < M_{i}, where f_{i}(j) is the ith component of the jth objective function value in the Pareto set, and M_{i} is an upper bound for the ith component for all points in the Pareto set. In this figure, M is called the Reference Point. The shades of gray in the figure denote portions of the volume that some calculation algorithms use as part of an inclusionexclusion calculation. For details, see Fleischer [3].
Volume change is one factor in stopping the algorithm. For details, see Stopping Conditions. 
Distance  Distance is a measure of the closeness of an individual to
its nearest neighbors. The The algorithm sets the distance of
individuals at the extreme positions to
The algorithm sorts each dimension separately, so the term neighbors means neighbors in each dimension. Individuals of the same rank with a higher distance have a higher chance of selection (higher distance is better). Distance is one factor in the calculation of the spread, which is part of a stopping criterion. For details, see Stopping Conditions. 
Spread  Spread is a measure of the movement of the Pareto set. To
calculate the spread, the
The spread is small when the extreme objective function values do not change much between iterations (that is, μ is small) and when the points on the Pareto front are spread evenly (that is, σ is small).

ParetoSetChangeTolerance  Stopping condition for the search.
paretosearch stops when the volume, spread,
or distance does not change by more than
ParetoSetChangeTolerance over a window of
iterations. For details, see Stopping Conditions. 
MinPollFraction  Minimum fraction of locations to poll during an iteration.
This option does not apply when the

Sketch of paretosearch
Algorithm
Initialize Search
To create the initial set of points, paretosearch
generates
options.ParetoSetSize
points from a quasirandom sample based
on the problem bounds, by default. For details, see Bratley and Fox [2]. When the problem
has over 500 dimensions, paretosearch
uses Latin
hypercube sampling to generate the initial points.
If a component has no bounds, paretosearch
uses an artificial
lower bound of 10
and an artificial upper bound of
10
.
If a component has only one bound, paretosearch
uses that
bound as an endpoint of an interval of width 20 + 2*abs(bound)
.
For example, if there is no upper bound for a component and there is a lower bound
of 15, paretosearch
uses an interval width of 20 + 2*15 = 55,
so uses an artificial upper bound of 15 + 55 = 70.
If you pass some initial points in options.InitialPoints
, then
paretosearch
uses those points as the initial points.
paretosearch
generates more points, if necessary, to obtain
at least options.ParetoSetSize
initial points.
paretosearch
then checks the initial points to ensure that
they are feasible with respect to the bounds and linear constraints. If necessary,
paretosearch
projects the initial points onto the linear
subspace of linearly feasible points by solving a linear programming problem. This
process can cause some points to coincide, in which case
paretosearch
removes any duplicate points.
paretosearch
does not alter initial points for artificial
bounds, only for specified bounds and linear constraints.
After moving the points to satisfy linear constraints, if necessary,
paretosearch
checks whether the points satisfy the
nonlinear constraints. paretosearch
gives a penalty value of
Inf
to any point that does not satisfy all nonlinear
constraints. Then paretosearch
calculates any missing objective
function values of the remaining feasible points.
Note
Currently, paretosearch
does not support nonlinear equality
constraints ceq(x) = 0
.
Create Archive and Incumbents
paretosearch
maintains two sets of points:
archive
— A structure that contains nondominated points associated with a mesh size belowoptions.MeshTolerance
and satisfying all constraints to withinoptions.ConstraintTolerance
. Thearchive
structure contains no more than2*options.ParetoSetSize
points and is initially empty. Each point inarchive
contains an associated mesh size, which is the mesh size at which the point was generated.iterates
— A structure containing nondominated points and possibly some dominated points associated with larger mesh sizes or infeasibility. Each point initerates
contains an associated mesh size.iterates
contains no more thanoptions.ParetoSetSize
points.
Poll to Find Better Points
paretosearch
polls points from iterates
,
with the polled points inheriting the associated mesh size from the point in
iterates
. The paretosearch
algorithm
uses a poll that maintains feasibility with respect to bounds and all linear
constraints.
If the problem has nonlinear constraints, paretosearch
computes the feasibility of each poll point. paretosearch
keeps
the score of infeasible points separately from the score of feasible points. The
score of a feasible point is the vector of objective function values of that point.
The score of an infeasible point is the sum of the nonlinear infeasibilities.
paretosearch
polls at least
MinPollFraction
*(number of points in pattern) locations for
each point in iterates
. If the polled points give at least one
nondominated point with respect to the incumbent (original) point, the poll is
considered a success. Otherwise, paretosearch
continues to poll
until it either finds a nondominated point or runs out of points in the pattern. If
paretosearch
runs out of points and does not produce a
nondominated point, paretosearch
declares the poll unsuccessful
and halves the mesh size.
If the poll finds nondominated points, paretosearch
extends
the poll in the successful directions repeatedly, doubling the mesh size each time,
until the extension produces a dominated point. During this extension, if the mesh
size exceeds options.MaxMeshSize
(default value:
Inf
), the poll stops. If the objective function values
decrease to Inf
, paretosearch
declares the
problem unbounded and stops.
Update archive
and iterates
Structures
After polling all the points in iterates
, the algorithm
examines the new points together with the points in the iterates
and archive
structures. paretosearch
computes the rank, or Pareto front number, of each point and then does the
following.
Mark for removal all points that do not have rank 1 in
archive
.Mark new rank
1
points for insertion intoiterates
.Mark feasible points in
iterates
whose associated mesh size is less thanoptions.MeshTolerance
for transfer toarchive
.Mark dominated points in
iterates
for removal only if they prevent new nondominated points from being added toiterates
.
paretosearch
then computes the volume and distance measures
for each point. If archive
will overflow as a result of marked
points being included, then the points with the largest volume occupy
archive
, and the others leave. Similarly, the new points
marked for addition to iterates
enter iterates
in order of their volumes.
If iterates
is full and has no dominated points, then
paretosearch
adds no points to iterates
and declares the iteration to be unsuccessful. paretosearch
multiplies the mesh sizes in iterates
by 1/2.
Stopping Conditions
For three or fewer objective functions, paretosearch
uses
volume and spread as stopping measures. For four or more objectives,
paretosearch
uses distance and spread as stopping measures.
In the remainder of this discussion, the two measures that
paretosearch
uses are denoted the
applicable measures.
The algorithm maintains vectors of the last eight values of the applicable
measures. After eight iterations, the algorithm checks the values of the two
applicable measures at the beginning of each iteration, where tol =
options.ParetoSetChangeTolerance
:
spreadConverged = abs(spread(end  1)  spread(end)) <= tol*max(1,spread(end  1));
volumeConverged = abs(volume(end  1)  volume(end)) <= tol*max(1,volume(end  1));
distanceConverged = abs(distance(end  1)  distance(end)) <= tol*max(1,distance(end  1));
If either applicable test is true
, the algorithm stops.
Otherwise, the algorithm computes the max of squared terms of the Fourier transforms
of the applicable measures minus the first term. The algorithm then compares the
maxima to their deleted terms (the DC components of the transforms). If either
deleted term is larger than 100*tol*(max of all other terms)
,
then the algorithm stops. This test essentially determines that the sequence of
measures is not fluctuating, and therefore has converged.
Additionally, a plot function or output function can stop the algorithm, or the algorithm can stop because it exceeds a time limit or function evaluation limit.
Returned Values
The algorithm returns the points on the Pareto front as follows.
paretosearch
combines the points inarchive
anditerates
into one set.When there are three or fewer objective functions,
paretosearch
returns the points from the largest volume to the smallest, up to at mostParetoSetSize
points.When there are four or more objective functions,
paretosearch
returns the points from the largest distance to the smallest, up to at mostParetoSetSize
points.
Modifications for Parallel Computation and Vectorized Function Evaluation
When paretosearch
computes objective function values in
parallel or in a vectorized fashion (UseParallel
is
true
or UseVectorized
is
true
), there are some changes to the algorithm.
When
UseVectorized
istrue
,paretosearch
ignores theMinPollFraction
option and evaluates all poll points in the pattern.When computing in parallel,
paretosearch
sequentially examines each point initerates
and performs a parallel poll from each point. After returningMinPollFraction
fraction of the poll points,paretosearch
determines if any poll points dominate the base point. If so, the poll is deemed successful, and any other parallel evaluations halt. If not, polling continues until a dominating point appears or the poll is done.paretosearch
performs objective function evaluations either on workers or in a vectorized fashion, but not both. If you set bothUseParallel
andUseVectorized
totrue
,paretosearch
calculates objective function values in parallel on workers, but not in a vectorized fashion. In this case,paretosearch
ignores theMinPollFraction
option and evaluates all poll points in the pattern.
Run paretosearch
Quickly
The fastest way to run paretosearch
depends on several
factors.
If objective function evaluations are slow, then it is usually fastest to use parallel computing. The overhead in parallel computing can be substantial when objective function evaluations are fast, but when they are slow, it is usually best to use more computing power.
Note
Parallel computing requires a Parallel Computing Toolbox™ license.
If objective function evaluations are not very time consuming, then it is usually fastest to use vectorized evaluation. However, this is not always the case, because vectorized computations evaluate an entire pattern, whereas serial evaluations can take just a small fraction of a pattern. In high dimensions especially, this reduction in evaluations can cause serial evaluation to be faster for some problems.
To use vectorized computing, your objective function must accept a matrix with an arbitrary number of rows. Each row represents one point to evaluate. The objective function must return a matrix of objective function values with the same number of rows as it accepts, with one column for each objective function. For a singleobjective discussion, see Vectorize the Fitness Function (
ga
) or Vectorized Objective Function (patternsearch
).
References
[1] Custòdio, A. L., J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente. Direct Multisearch for Multiobjective Optimization. SIAM J. Optim., 21(3), 2011, pp. 1109–1140. Preprint available at https://www.researchgate.net/publication/220133323_Direct_Multisearch_for_Multiobjective_Optimization.
[2] Bratley, P., and B. L. Fox. Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Software 14, 1988, pp. 88–100.
[3] Fleischer, M. The Measure of Pareto Optima: Applications to MultiObjective Metaheuristics. In "Proceedings of the Second International Conference on Evolutionary MultiCriterion Optimization—EMO" April 2003 in Faro, Portugal. Published by SpringerVerlag in the Lecture Notes in Computer Science series, Vol. 2632, pp. 519–533. Preprint available at https://api.drum.lib.umd.edu/server/api/core/bitstreams/4241d9c0f5144a41bd5807ac2538d918/content.