A data scientist at a manufacturing firm is tasked with optimizing a production plan for two products to maximize profit. The production of each product requires specific, limited amounts of labor, raw materials, and machine time. The profit per unit for each product is constant. The data scientist correctly formulates this as a linear programming problem and selects the Simplex method for solving it. Which of the following statements accurately describes the core principle of how the Simplex method will find the optimal solution in this scenario?
It uses gradient descent to navigate the solution space and find the global maximum of the objective function.
It systematically evaluates the vertices of the feasible region, a convex polytope, to find the optimal solution.
It divides the problem into smaller, recursive subproblems and combines the solutions using dynamic programming.
It randomly samples points within the feasible region, using a probabilistic approach to converge on the optimal solution.
The correct answer is that the Simplex method systematically evaluates the vertices of the feasible region, which is a convex polytope, to find the optimal solution. For a linear programming problem, the optimal solution will always occur at a vertex (or corner point) of the feasible region defined by the constraints. The Simplex algorithm is a deterministic procedure that starts at a feasible vertex and iteratively moves to an adjacent vertex that improves the objective function's value, until an optimum is reached.
Gradient descent is an optimization technique used primarily for unconstrained problems or non-linear problems, often in training machine learning models. It follows the negative gradient of the objective function, which is not how the Simplex method operates on a linear problem with a constant gradient.
Dividing a problem into smaller, recursive subproblems is characteristic of dynamic programming, not the Simplex method.
Using random sampling to converge on a solution describes stochastic optimization methods, such as Monte Carlo techniques. The Simplex method is a deterministic, not a probabilistic, algorithm.
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What is a feasible region in linear programming?
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Why does the optimal solution in linear programming always occur at a vertex?