Constrained optimization in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is constrained optimization?

Constrained optimization is a way to find the best solution to a problem while following some rules or limits called constraints.

beginner

Name two types of constraints in optimization problems.

Two common types of constraints are:
1. Equality constraints (e.g., x + y = 10)
2. Inequality constraints (e.g., x ≥ 0)

beginner

Which function in scipy.optimize is commonly used for constrained optimization?

The function scipy.optimize.minimize is commonly used for constrained optimization by specifying constraints and bounds.

intermediate

How do you define constraints for scipy.optimize.minimize?

Constraints are defined as a list of dictionaries with keys like type ('eq' for equality, 'ineq' for inequality) and fun (a function that returns the constraint value).

intermediate

What does the 'method' parameter in scipy.optimize.minimize control?

The 'method' parameter chooses the algorithm to solve the optimization, such as 'SLSQP' which supports constraints and bounds.

What does an inequality constraint like 'x >= 0' mean in optimization?

A x must be exactly 0

B x must be greater than or equal to 0

C x must be less than 0

D x can be any value

An inequality constraint 'x >= 0' means x must be greater than or equal to zero.

Which scipy.optimize method supports constraints directly?

A Nelder-Mead

B CG

C SLSQP

D BFGS

The 'SLSQP' method supports both equality and inequality constraints.

In scipy.optimize.minimize, how are equality constraints specified?

A type='eq'

B type='ineq'

C type='equal'

D type='equal_to'

Equality constraints use type='eq' in the constraints dictionary.

What is the purpose of bounds in constrained optimization?

A To limit the range of variables

B To define the objective function

C To set the initial guess

D To choose the solver method

Bounds limit the range of values that variables can take.

What does the function passed to 'fun' in a constraint dictionary return?

A The initial guess

B The objective function value

C The gradient of the objective

D The constraint value (should be zero or positive)

The function returns the value of the constraint, which should be zero for equality or non-negative for inequality.

Explain how to set up a constrained optimization problem using scipy.optimize.minimize.

Think about the inputs needed for minimize to handle constraints.

Describe the difference between equality and inequality constraints with examples.

Consider how each constraint restricts possible solutions.

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