Constrained optimization in SciPy - Time & Space Complexity

Time Complexity: Constrained optimization

O(n^3)

Understanding Time Complexity

When solving constrained optimization problems with scipy, it is important to understand how the time needed grows as the problem size increases.

We want to know how the number of calculations changes when we add more variables or constraints.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

from scipy.optimize import minimize

# Objective function
fun = lambda x: (x[0]-1)**2 + (x[1]-2.5)**2

# Constraint: x0 + x1 >= 2
cons = ({'type': 'ineq', 'fun': lambda x: x[0] + x[1] - 2})

# Initial guess
x0 = [2, 0]

# Run constrained optimization
res = minimize(fun, x0, constraints=cons, method='SLSQP')

This code finds the minimum of a function with one inequality constraint using the SLSQP method.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Iterative calls to the objective and constraint functions during optimization steps.
• How many times: The optimizer runs multiple iterations, each evaluating the functions and gradients.

How Execution Grows With Input

As the number of variables and constraints grows, the optimizer must do more work each iteration.

Input Size (n variables)	Approx. Operations
10	Thousands of function and gradient evaluations
100	Millions of evaluations and matrix operations
1000	Billions of calculations, much slower convergence

Pattern observation: The work grows quickly because each iteration involves matrix calculations that scale roughly with the square or cube of the number of variables.

Final Time Complexity

Time Complexity: O(n^3)

This means the time to solve the problem grows roughly with the cube of the number of variables, due to matrix operations inside the optimizer.

Common Mistake

[X] Wrong: "Adding more constraints or variables only increases time a little bit."

[OK] Correct: Each new variable or constraint can greatly increase the size of matrices the optimizer handles, causing time to grow much faster than expected.

Interview Connect

Understanding how optimization time grows helps you explain trade-offs when solving real problems, showing you know what affects performance beyond just writing code.

Self-Check

"What if we switched from SLSQP to a gradient-free method? How would the time complexity change?"

Practice

1. What is the main purpose of constrained optimization in scipy.optimize.minimize?

easy

A. To only check if constraints are valid

B. To find any random solution without restrictions

C. To find the best solution while respecting given limits or rules

D. To maximize the function without any constraints

Solution

1. Step 1: Understand constrained optimization goal

   Constrained optimization aims to find the best (minimum or maximum) value of a function while following certain limits or rules called constraints.

2. Step 2: Relate to scipy.optimize.minimize usage

   The minimize function in scipy can include constraints and bounds to ensure the solution respects these limits.

3. Final Answer:

   To find the best solution while respecting given limits or rules -> Option C

4. Quick Check:

   Constrained optimization = best solution within limits [OK]

Hint: Constrained means limits must be followed [OK]

Common Mistakes:

• Thinking constraints are optional or ignored
• Confusing constrained with unconstrained optimization
• Assuming constraints only check validity, not guide solution

2. Which of the following is the correct way to define the inequality constraint `x[0] >= 1` for scipy.optimize.minimize?

easy

A. constraint = {'type': 'eq', 'fun': lambda x: x[0] - 1}

B. constraint = {'type': 'ineq', 'fun': lambda x: x[0] - 1}

C. constraint = {'type': 'ineq', 'fun': lambda x: 1 - x[0]}

D. constraint = {'type': 'eq', 'fun': lambda x: 1 - x[0]}

Solution

1. Step 1: Recall inequality constraint definition

   Inequality constraints require the function to be >= 0. So, fun(x) >= 0 means the constraint holds.

2. Step 2: Check each option

   constraint = {'type': 'ineq', 'fun': lambda x: x[0] - 1}: lambda x: x[0] - 1 means x[0] - 1 >= 0 or x[0] >= 1, correct for the constraint.
   Options B and D use 'eq' which means equality, not inequality.
   constraint = {'type': 'ineq', 'fun': lambda x: 1 - x[0]} means 1 - x[0] >= 0 or x[0] <= 1, which is a different constraint.

3. Final Answer:

   constraint = {'type': 'ineq', 'fun': lambda x: x[0] - 1} -> Option B

4. Quick Check:

   Inequality means fun(x) >= 0 [OK]

Hint: Inequality means function output ≥ 0 [OK]

Common Mistakes:

• Using 'eq' instead of 'ineq' for inequalities
• Reversing the inequality sign in the function
• Confusing the function output sign for constraints

3. What will be the output of this code snippet?

from scipy.optimize import minimize

fun = lambda x: (x[0]-2)**2 + (x[1]-3)**2
cons = ({'type': 'eq', 'fun': lambda x: x[0] + x[1] - 5})
res = minimize(fun, [0, 0], constraints=cons)
print(res.x.round(2))

medium

A. [2.5 2.5]

B. [1.0 4.0]

C. [0.0 5.0]

D. [2.0 3.0]

Solution

1. Step 1: Understand the objective and constraint

   The function minimizes distance from point (2,3). The constraint requires x[0] + x[1] = 5.

2. Step 2: Find point on line x0+x1=5 closest to (2,3)

   We want to minimize (x0-2)^2 + (x1-3)^2 with x0+x1=5. Using substitution: x1=5-x0.
   Objective becomes (x0-2)^2 + (5-x0-3)^2 = (x0-2)^2 + (2 - x0)^2.
   Expanding: (x0-2)^2 + (2-x0)^2 = 2*(x0-2)^2.
   Minimum at x0=2, then x1=3, which satisfies the constraint.

3. Step 3: Check options

   [2.0 3.0] matches the minimum (2.0, 3.0). [1.0 4.0] sums to 5 but is farther from (2,3). The solver converges to [2.0, 3.0].

4. Final Answer:

   [2.0 3.0] -> Option D

5. Quick Check:

   Minimum at (2,3) on constraint line [OK]

Hint: Check constraint line and closest point to target [OK]

Common Mistakes:

• Ignoring the constraint when finding minimum
• Confusing initial guess with solution
• Assuming solver returns initial guess

4. Identify the error in this constrained optimization code:

from scipy.optimize import minimize

fun = lambda x: x[0]**2 + x[1]**2
cons = [{'type': 'ineq', 'fun': lambda x: x[0] + x[1] - 1}]
res = minimize(fun, [0, 0], constraints=cons)
print(res.x)

medium

A. Initial guess violates the inequality constraint

B. Constraint function should return negative values

C. Constraints must be a dictionary, not a list

D. Objective function must be linear

Solution

1. Step 1: Check initial guess against constraint

   The inequality constraint requires x[0] + x[1] - 1 >= 0, or x[0] + x[1] >= 1.
   The initial guess is [0,0], sum is 0, which violates the constraint.

2. Step 2: Understand impact of violating initial guess

   Starting from a point that violates constraints can cause solver to fail or give warnings.
   Other options are incorrect because:
   - Constraint function should return >=0 for valid points.
   - Constraints can be a list of dicts.
   - Objective function can be nonlinear.

3. Final Answer:

   Initial guess violates the inequality constraint -> Option A

4. Quick Check:

   Initial guess must satisfy constraints [OK]

Hint: Check if initial guess meets constraints [OK]

Common Mistakes:

• Ignoring initial guess validity
• Misunderstanding constraint function sign
• Thinking constraints must be single dict

5. You want to minimize f(x) = (x[0]-1)^2 + (x[1]-2)^2 with constraints x[0] >= 0, x[1] >= 0, and x[0] + 2*x[1] = 3. Which setup correctly applies these constraints using scipy.optimize.minimize?

hard

A. bounds = [(0, None), (0, None)]; cons = {'type': 'eq', 'fun': lambda x: x[0] + 2*x[1] - 3}

B. bounds = [(None, 0), (None, 0)]; cons = {'type': 'ineq', 'fun': lambda x: 3 - x[0] - 2*x[1]}

C. bounds = [(0, None), (0, None)]; cons = {'type': 'ineq', 'fun': lambda x: x[0] + 2*x[1] - 3}

D. bounds = [(0, None), (0, None)]; cons = {'type': 'ineq', 'fun': lambda x: 3 - x[0] - 2*x[1]}

Solution

1. Step 1: Set bounds for inequalities

   Constraints x[0] >= 0 and x[1] >= 0 are bounds: [(0, None), (0, None)] means lower bound 0, no upper bound.

2. Step 2: Define equality constraint

   The equality x[0] + 2*x[1] = 3 is expressed as fun(x) = x[0] + 2*x[1] - 3 = 0 with type 'eq'.

3. Step 3: Check options

   bounds = [(0, None), (0, None)]; cons = {'type': 'eq', 'fun': lambda x: x[0] + 2*x[1] - 3} correctly sets bounds and equality constraint.
   bounds = [(None, 0), (None, 0)]; cons = {'type': 'ineq', 'fun': lambda x: 3 - x[0] - 2*x[1]} reverses bounds and uses inequality instead of equality.
   bounds = [(0, None), (0, None)]; cons = {'type': 'ineq', 'fun': lambda x: x[0] + 2*x[1] - 3} uses inequality for equality constraint.
   bounds = [(0, None), (0, None)]; cons = {'type': 'ineq', 'fun': lambda x: 3 - x[0] - 2*x[1]} uses inequality (x[0] + 2*x[1] <= 3) instead of equality.

4. Final Answer:

   bounds = [(0, None), (0, None)]; cons = {'type': 'eq', 'fun': lambda x: x[0] + 2*x[1] - 3} -> Option A

5. Quick Check:

   Bounds for inequalities + 'eq' for equality [OK]

Hint: Use bounds for ≥ constraints, 'eq' for equalities [OK]

Common Mistakes:

• Using 'ineq' for equality constraints
• Reversing bounds limits
• Confusing constraint function sign