Optimization assigns values to decision variables to find the best available solution to a problem. An objective function measures what “best” means, such as minimum cost or maximum profit. Constraints limit which assignments are acceptable. Searching for the best solution therefore means comparing candidates within a feasible search space, though a practical algorithm may settle for a near-optimal result. section 1.3 section 1.6
To formulate a problem, identify:
- Decision variables: the inputs you can control, such as a ticket price or a selection of parcels. section 1.6 section 1.3.2
- Objective functions: the quantities to minimize or maximize. A problem may have one objective or several competing objectives. section 1.3.2
- Constraints: bounds and other restrictions. Hard constraints must hold; soft constraints express preferences and can be represented through rewards or penalties in the objective. section 1.3.3
Pricing one ticket
Suppose ticket price x must lie between $0 and $250, and profit is f(x) = −20x² + 6,200x − 350,000. Setting the derivative, −40x + 6,200, to zero gives x = $155 and profit of $130,500. This is a small problem with one decision variable and one objective; larger problems may require a search algorithm rather than a hand calculation. section 1.3.2 section 1.2
Constraints can change how candidates are judged. In cargo-bike loading, each parcel has a weight and profit, and an overweight load incurs a penalty. In timetabling, two lectures cannot occupy the same room at once, while placing consecutive lectures in nearby rooms is a preference. First decide which restrictions are inviolable; then decide how to score any preferences. section 1.3.3
An optimizer searches among candidate assignments. Exploration tests new regions; exploitation concentrates on a region that has already produced good results. Too much exploitation can leave the search near a local optimum, while too much exploration costs time. The required turnaround matters: a design decision may permit days of computation, whereas a repeatedly solved control problem may allow only milliseconds or seconds. The benefit must also justify implementation costs; elaborate routing changes may offer little value for a handful of deliveries. section 1.5 section 1.2 section 1.1
For a continuous objective, simulated annealing can generate neighboring values and sometimes accept a worse one while its temperature decreases. On f(x) = (x − 6)² with 0 ≤ x ≤ 31, a hand iteration accepts an improvement from x = 2 to x = 2.25, then rejects a worse candidate according to its acceptance probability. Python implementations can scale the experiment: define the objective and bounds, run the solver, inspect its result, and tune its parameters. Tabu search offers another approach to continuous function optimization; parameter tuning affects whether it finds an optimal or near-optimal solution. section 5.3 section 6.4
Listing 5.1 Function optimization using scipy.optimize.dual_annealing
#!pip install scipy
import numpy as np
from scipy.optimize import dual_annealing
def objective_function(solution): #1
return solution[0]**2 +2*(solution[1]**2) - 0.3*np.cos(3*np.pi*solution[0]) –
0.4*np.cos(4*np.pi*solution[1]) + 0.7
bounds = np.asarray([[-100, 100], [-100, 100]]) #2
res_dual = dual_annealing(objective_function, bounds=bounds, maxiter = 100) #3
print('Dual Annealing Solution: f(%s) = %.5f' % (res_dual['x'], res_dual['fun'])) #4
#1 Define the objective function or functions (e.g., Bohachevsky function).
#2 Define the boundary constraints of the decision variables.
#3 Perform the dual annealing search.
#4 Print the dual annealing solution.
This Python listing shows the practical interface: define an objective, set variable bounds, run SciPy’s dual_annealing, and print the candidate solution and its objective value. section 5.3
