Vector Variables

Working with high-dimensional decision variables

Published

February 8, 2026

1 Overview

VectorVariable allows you to define and manipulate arrays of decision variables efficiently. Instead of creating variables one by one in a loop, you can create thousands at once and perform vectorized operations.

from optyx import VectorVariable

# Create 100 variables: x[0], x[1], ..., x[99]
x = VectorVariable("x", 100, lb=0)

2 Creating Vectors

2.1 Basic Creation

# Unbounded vector of size 10
x = VectorVariable("x", 10)

# Non-negative vector
y = VectorVariable("y", 10, lb=0)

# Bounded vector
z = VectorVariable("z", 10, lb=-1, ub=1)

# Binary vector (relaxed to [0, 1] in standard solvers)
b = VectorVariable("b", 10, domain="binary")

2.2 From NumPy Arrays

You can create a vector matching the size of an existing data array:

import numpy as np
data = np.array([1.5, 2.0, 3.5])
x = VectorVariable.from_numpy("x", data, lb=0)
# x has size 3

3 Indexing and Slicing

VectorVariable supports standard Python indexing and slicing:

x = VectorVariable("x", 10)

# Access single element (returns Variable)
x0 = x[0]

# Slice (returns new VectorVariable)
subset = x[2:5]  # x[2], x[3], x[4]

# Negative indexing
last = x[-1]

4 Vector Operations

4.1 Arithmetic

You can perform element-wise arithmetic with scalars or other vectors:

x = VectorVariable("x", 3)
y = VectorVariable("y", 3)

# Element-wise addition
z = x + y

# Scalar multiplication
w = 2 * x

# Linear combination
expr = 2*x + 3*y + 1

4.2 Dot Product

Compute the dot product between two vectors or a vector and a constant array:

# Vector-Vector dot product
d = x.dot(y)  # or x @ y

# Vector-Constant dot product
c = np.array([1, 2, 3])
val = c @ x   # 1*x[0] + 2*x[1] + 3*x[2]

4.3 Norms

Compute L1 and L2 norms:

from optyx import L1Norm, L2Norm

# L1 Norm: sum(|x_i|)
n1 = L1Norm(x)

# L2 Norm: sqrt(sum(x_i^2))
n2 = L2Norm(x)

4.4 Summation

Sum all elements in a vector:

from optyx import sum

total = sum(x)

5 Constraints

Constraints can be applied to entire vectors at once. This creates a list of constraints, one for each element.

x = VectorVariable("x", 10)
y = VectorVariable("y", 10)

# Element-wise constraints
prob.subject_to(x >= 0)      # x[i] >= 0 for all i
prob.subject_to(x + y <= 1)  # x[i] + y[i] <= 1 for all i
prob.subject_to(x <= y)      # x[i] <= y[i] for all i

6 Integration with NumPy

VectorVariable is designed to work seamlessly with NumPy arrays for data-driven optimization.

import numpy as np

# Data
prices = np.array([10, 20, 30])
costs = np.array([5, 15, 25])
demand = np.array([100, 200, 150])

# Variables
qty = VectorVariable("qty", 3, lb=0)

# Objective: Maximize profit
revenue = prices @ qty
cost = costs @ qty
prob.maximize(revenue - cost)

# Constraints
prob.subject_to(qty <= demand)