Bounds and constraints in SciPy - Step-by-Step Execution

Concept Flow - Bounds and constraints

Define objective function

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Set bounds for variables

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Set constraints (eq/ineq)

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Call optimizer with function, bounds, constraints

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Optimizer iterates to find solution

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Check if solution meets bounds and constraints

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Return optimal solution or failure

We define the function to minimize, set variable limits (bounds), add rules (constraints), then run the optimizer to find the best solution within those limits.

Execution Sample

SciPy

from scipy.optimize import minimize

# Objective function
def f(x):
    return (x[0]-1)**2 + (x[1]-2.5)**2

# Bounds and constraints
bounds = [(0, None), (0, None)]
cons = {'type': 'ineq', 'fun': lambda x: x[0] - 1}

res = minimize(f, [2, 0], bounds=bounds, constraints=[cons])
print(res.x)

This code finds the minimum of a function with bounds and a constraint on x[0].

Execution Table

Step	x values	Objective f(x)	Constraint x[0]-1 >= 0	Bounds check	Action
1	[2, 0]	7.25	1 >= 0 True	x[0]=2>=0, x[1]=0>=0 True	Start at initial guess
2	[1.5, 1]	2.5	0.5 >= 0 True	Bounds OK	Optimizer tries better x
3	[1.0, 2.0]	0.25	0 >= 0 True	Bounds OK	Improved solution
4	[1.0, 2.5]	0.0	0 >= 0 True	Bounds OK	Found minimum satisfying constraints
5	[0.9, 2.5]	0.01	-0.1 >= 0 False	Bounds OK	Constraint violated, discard
Exit	-	-	-	-	Optimization stops: solution found within bounds and constraints

💡 Optimization stops because the solution satisfies all bounds and constraints.

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
x[0]	2	1.5	1.0	1.0	0.9	1.0
x[1]	0	1	2.0	2.5	2.5	2.5
f(x)	7.25	2.5	0.25	0.0	0.01	0.0
Constraint (x[0]-1)	1	0.5	0	0	-0.1	0

Key Moments - 2 Insights

Why does the optimizer reject the point [0.9, 2.5] even though it has a low objective value?

Because the constraint x[0] - 1 >= 0 is violated at this point (value is -0.1), so the optimizer discards it as shown in step 5 of the execution table.

What does the bounds parameter do in the optimization?

Bounds restrict variables to stay within specified limits (here x[0] and x[1] must be >= 0). The optimizer checks these at each step (see 'Bounds check' column).

Visual Quiz - 3 Questions

Look at the execution table, what is the value of the constraint at step 3?

A 0

B -0.1

C 0.5

D 1

💡 Hint

Check the 'Constraint x[0]-1 >= 0' column at step 3 in the execution table.

At which step does the optimizer find a solution that satisfies all constraints and bounds?

A Step 2

B Step 4

C Step 3

D Step 5

💡 Hint

Look for the step where 'Constraint' and 'Bounds check' are True and objective is minimal.

If the bounds were changed to allow negative values, what would happen to the solution?

A Constraints would be ignored

B Solution would stay the same

C Solution might move to negative x values

D Optimization would fail

💡 Hint

Refer to variable_tracker and how bounds limit variable values.

Concept Snapshot

Bounds and constraints guide optimization:
- Bounds limit variable ranges (e.g., x >= 0)
- Constraints add rules (e.g., x[0] - 1 >= 0)
- Optimizer searches for minimum within these limits
- Violations discard candidate solutions
- Use scipy.optimize.minimize with bounds and constraints parameters

Full Transcript

In optimization, we want to find the best values for variables that minimize a function. Sometimes, variables must stay within limits called bounds, like being non-negative. Also, there can be rules called constraints that the solution must follow. We define the function, set bounds and constraints, then run the optimizer. It tries different values step by step, checking if they meet bounds and constraints. If a candidate breaks a rule, it is rejected. The process continues until the best valid solution is found. This example shows how the optimizer moves from an initial guess to the final solution, respecting all limits.