optimeq
Create empty optimization equality array
Syntax
Description
Use optimeq
to initialize a set of equality
expressions.
Tip
For the full workflow, see Problem-Based Optimization Workflow or Problem-Based Workflow for Solving Equations.
Examples
Create Equality Constraints in Loop
Create equality constraints for an inventory model. The stock of goods at the start of each period is equal to the stock at the end of the previous period. During each period, the stock increases by buy
and decreases by sell
. The variable stock
is the stock at the end of the period.
N = 12; stock = optimvar('stock',N,1,'Type','integer','LowerBound',0); buy = optimvar('buy',N,1,'Type','integer','LowerBound',0); sell = optimvar('sell',N,1,'Type','integer','LowerBound',0); initialstock = 100; stockbalance = optimeq(N,1); for t = 1:N if t == 1 enterstock = initialstock; else enterstock = stock(t-1); end stockbalance(t) = stock(t) == enterstock + buy(t) - sell(t); end show(stockbalance)
(1, 1) -buy(1) + sell(1) + stock(1) == 100 (2, 1) -buy(2) + sell(2) - stock(1) + stock(2) == 0 (3, 1) -buy(3) + sell(3) - stock(2) + stock(3) == 0 (4, 1) -buy(4) + sell(4) - stock(3) + stock(4) == 0 (5, 1) -buy(5) + sell(5) - stock(4) + stock(5) == 0 (6, 1) -buy(6) + sell(6) - stock(5) + stock(6) == 0 (7, 1) -buy(7) + sell(7) - stock(6) + stock(7) == 0 (8, 1) -buy(8) + sell(8) - stock(7) + stock(8) == 0 (9, 1) -buy(9) + sell(9) - stock(8) + stock(9) == 0 (10, 1) -buy(10) + sell(10) - stock(9) + stock(10) == 0 (11, 1) -buy(11) + sell(11) - stock(10) + stock(11) == 0 (12, 1) -buy(12) + sell(12) - stock(11) + stock(12) == 0
Include the constraints in an optimization problem.
prob = optimproblem; prob.Constraints.stockbalance = stockbalance;
Instead of using a loop, you can create the same constraints by using matrix operations on the variables.
stockbalance2 = optimeq(12, 1); t = 2:12; stockbalance2(t) = stock(t) == stock(t-1) + buy(t) - sell(t); stockbalance2(1) = stock(1) == initialstock + buy(1) - sell(1);
Display the new constraints. Note that they are the same as the constraints in stockbalance
.
show(stockbalance2)
(1, 1) -buy(1) + sell(1) + stock(1) == 100 (2, 1) -buy(2) + sell(2) - stock(1) + stock(2) == 0 (3, 1) -buy(3) + sell(3) - stock(2) + stock(3) == 0 (4, 1) -buy(4) + sell(4) - stock(3) + stock(4) == 0 (5, 1) -buy(5) + sell(5) - stock(4) + stock(5) == 0 (6, 1) -buy(6) + sell(6) - stock(5) + stock(6) == 0 (7, 1) -buy(7) + sell(7) - stock(6) + stock(7) == 0 (8, 1) -buy(8) + sell(8) - stock(7) + stock(8) == 0 (9, 1) -buy(9) + sell(9) - stock(8) + stock(9) == 0 (10, 1) -buy(10) + sell(10) - stock(9) + stock(10) == 0 (11, 1) -buy(11) + sell(11) - stock(10) + stock(11) == 0 (12, 1) -buy(12) + sell(12) - stock(11) + stock(12) == 0
Creating constraints in a loop can be more time consuming than creating constraints by using matrix operations.
Create Indexed Equalities in Loop
Create indexed equalities for a problem that involves shipping goods between airports. First, create indices representing airports.
airports = ["LAX" "JFK" "ORD"];
Create indices representing goods to be shipped from one airport to another.
goods = ["Electronics" "Foodstuffs" "Clothing" "Raw Materials"];
Create an array giving the weight of each unit of the goods.
weights = [1 20 5 100];
Create a variable array representing quantities of goods to be shipped from each airport to each other airport. quantities(airport1,airport2,goods)
represents the quantity of goods
being shipped from airport1
to airport2
.
quantities = optimvar('quantities',airports,airports,goods,'LowerBound',0);
Create an equality constraint that the sum of the weights of goods being shipped from each airport is equal to the sum of the weights of goods being shipped to the airport.
eq = optimeq(airports); outweight = optimexpr(size(eq)); inweight = optimexpr(size(eq)); for i = 1:length(airports) temp = optimexpr; temp2 = optimexpr; for j = 1:length(airports) for k = 1:length(goods) temp = temp + quantities(i,j,k)*weights(k); temp2 = temp2 + quantities(j,i,k)*weights(k); end end outweight(i) = temp; inweight(i) = temp2; eq(i) = outweight(i) == inweight(i); end
Examine the equalities.
show(eq)
(1, 'LAX') -quantities('JFK', 'LAX', 'Electronics') - quantities('ORD', 'LAX', 'Electronics') + quantities('LAX', 'JFK', 'Electronics') + quantities('LAX', 'ORD', 'Electronics') - 20*quantities('JFK', 'LAX', 'Foodstuffs') - 20*quantities('ORD', 'LAX', 'Foodstuffs') + 20*quantities('LAX', 'JFK', 'Foodstuffs') + 20*quantities('LAX', 'ORD', 'Foodstuffs') - 5*quantities('JFK', 'LAX', 'Clothing') - 5*quantities('ORD', 'LAX', 'Clothing') + 5*quantities('LAX', 'JFK', 'Clothing') + 5*quantities('LAX', 'ORD', 'Clothing') - 100*quantities('JFK', 'LAX', 'Raw Materials') - 100*quantities('ORD', 'LAX', 'Raw Materials') + 100*quantities('LAX', 'JFK', 'Raw Materials') + 100*quantities('LAX', 'ORD', 'Raw Materials') == 0 (1, 'JFK') quantities('JFK', 'LAX', 'Electronics') - quantities('LAX', 'JFK', 'Electronics') - quantities('ORD', 'JFK', 'Electronics') + quantities('JFK', 'ORD', 'Electronics') + 20*quantities('JFK', 'LAX', 'Foodstuffs') - 20*quantities('LAX', 'JFK', 'Foodstuffs') - 20*quantities('ORD', 'JFK', 'Foodstuffs') + 20*quantities('JFK', 'ORD', 'Foodstuffs') + 5*quantities('JFK', 'LAX', 'Clothing') - 5*quantities('LAX', 'JFK', 'Clothing') - 5*quantities('ORD', 'JFK', 'Clothing') + 5*quantities('JFK', 'ORD', 'Clothing') + 100*quantities('JFK', 'LAX', 'Raw Materials') - 100*quantities('LAX', 'JFK', 'Raw Materials') - 100*quantities('ORD', 'JFK', 'Raw Materials') + 100*quantities('JFK', 'ORD', 'Raw Materials') == 0 (1, 'ORD') quantities('ORD', 'LAX', 'Electronics') + quantities('ORD', 'JFK', 'Electronics') - quantities('LAX', 'ORD', 'Electronics') - quantities('JFK', 'ORD', 'Electronics') + 20*quantities('ORD', 'LAX', 'Foodstuffs') + 20*quantities('ORD', 'JFK', 'Foodstuffs') - 20*quantities('LAX', 'ORD', 'Foodstuffs') - 20*quantities('JFK', 'ORD', 'Foodstuffs') + 5*quantities('ORD', 'LAX', 'Clothing') + 5*quantities('ORD', 'JFK', 'Clothing') - 5*quantities('LAX', 'ORD', 'Clothing') - 5*quantities('JFK', 'ORD', 'Clothing') + 100*quantities('ORD', 'LAX', 'Raw Materials') + 100*quantities('ORD', 'JFK', 'Raw Materials') - 100*quantities('LAX', 'ORD', 'Raw Materials') - 100*quantities('JFK', 'ORD', 'Raw Materials') == 0
To avoid the nested for
loops, express the equalities using standard MATLAB® operators. Create the array of departing quantities by summing over the arrival airport indices. Squeeze the result to remove the singleton dimension.
departing = squeeze(sum(quantities,2));
Calculate the weights of the departing quantities.
departweights = departing * weights';
Similarly, calculate the weights of arriving quantities.
arriving = squeeze(sum(quantities,1)); arriveweights = arriving*weights';
Create the constraints that the departing weights equal the arriving weights.
eq2 = departweights == arriveweights;
Include the appropriate index names for the equalities by setting the IndexNames
property.
eq2.IndexNames = {airports,{}};
Display the new equalities. Note that they match the previous equalities, but are transposed vectors.
show(eq2)
('LAX', 1) -quantities('JFK', 'LAX', 'Electronics') - quantities('ORD', 'LAX', 'Electronics') + quantities('LAX', 'JFK', 'Electronics') + quantities('LAX', 'ORD', 'Electronics') - 20*quantities('JFK', 'LAX', 'Foodstuffs') - 20*quantities('ORD', 'LAX', 'Foodstuffs') + 20*quantities('LAX', 'JFK', 'Foodstuffs') + 20*quantities('LAX', 'ORD', 'Foodstuffs') - 5*quantities('JFK', 'LAX', 'Clothing') - 5*quantities('ORD', 'LAX', 'Clothing') + 5*quantities('LAX', 'JFK', 'Clothing') + 5*quantities('LAX', 'ORD', 'Clothing') - 100*quantities('JFK', 'LAX', 'Raw Materials') - 100*quantities('ORD', 'LAX', 'Raw Materials') + 100*quantities('LAX', 'JFK', 'Raw Materials') + 100*quantities('LAX', 'ORD', 'Raw Materials') == 0 ('JFK', 1) quantities('JFK', 'LAX', 'Electronics') - quantities('LAX', 'JFK', 'Electronics') - quantities('ORD', 'JFK', 'Electronics') + quantities('JFK', 'ORD', 'Electronics') + 20*quantities('JFK', 'LAX', 'Foodstuffs') - 20*quantities('LAX', 'JFK', 'Foodstuffs') - 20*quantities('ORD', 'JFK', 'Foodstuffs') + 20*quantities('JFK', 'ORD', 'Foodstuffs') + 5*quantities('JFK', 'LAX', 'Clothing') - 5*quantities('LAX', 'JFK', 'Clothing') - 5*quantities('ORD', 'JFK', 'Clothing') + 5*quantities('JFK', 'ORD', 'Clothing') + 100*quantities('JFK', 'LAX', 'Raw Materials') - 100*quantities('LAX', 'JFK', 'Raw Materials') - 100*quantities('ORD', 'JFK', 'Raw Materials') + 100*quantities('JFK', 'ORD', 'Raw Materials') == 0 ('ORD', 1) quantities('ORD', 'LAX', 'Electronics') + quantities('ORD', 'JFK', 'Electronics') - quantities('LAX', 'ORD', 'Electronics') - quantities('JFK', 'ORD', 'Electronics') + 20*quantities('ORD', 'LAX', 'Foodstuffs') + 20*quantities('ORD', 'JFK', 'Foodstuffs') - 20*quantities('LAX', 'ORD', 'Foodstuffs') - 20*quantities('JFK', 'ORD', 'Foodstuffs') + 5*quantities('ORD', 'LAX', 'Clothing') + 5*quantities('ORD', 'JFK', 'Clothing') - 5*quantities('LAX', 'ORD', 'Clothing') - 5*quantities('JFK', 'ORD', 'Clothing') + 100*quantities('ORD', 'LAX', 'Raw Materials') + 100*quantities('ORD', 'JFK', 'Raw Materials') - 100*quantities('LAX', 'ORD', 'Raw Materials') - 100*quantities('JFK', 'ORD', 'Raw Materials') == 0
Creating constraints in a loop can be more time consuming than creating constraints by using matrix operations.
Input Arguments
N
— Size of constraint dimension
positive integer
Size of the constraint dimension, specified as a positive integer.
The size of
constr = optimeq(N)
isN
-by-1.The size of
constr = optimeq(N1,N2)
isN1
-by-N2
.The size of
constr = optimeq(N1,N2,...,Nk)
isN1
-by-N2
-by-...-by-Nk
.
Example: 5
Data Types: double
cstr
— Names for indexing
cell array of character vectors | string vector
Names for indexing, specified as a cell array of character vectors or a string vector.
Note
cstr
cannot be a string scalar such as "Tp"
,
but must be a vector such as ["Tp" "ul"]
. To specify a single
name, use {'Tp'}
or the equivalent
cellstr("Tp")
.
Example: {'red','orange','green','blue'}
Example: ["red";"orange";"green";"blue"]
Data Types: string
| cell
Output Arguments
eq
— Equalities
empty OptimizationEquality
array
Equalities, returned as an empty OptimizationEquality
array. Use eq
to initialize a loop
that creates equalities.
For example:
x = optimvar('x',8); eq = optimeq(4); for k = 1:4 eq(k) = 5*k*(x(2*k) - x(2*k-1)) == 10 - 2*k; end
Tips
You can use
optimconstr
instead ofoptimeq
to create equality constraints for optimization problems or equations for equation problems.
Version History
Introduced in R2019b
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