Gradient Descent and Linear Programming

The video explores the geometrical interpretation of linear programs, starting with a two-dimensional example. It explains how constraints form a feasible region, represented as a polygon, and how the objective function is minimized within this region. The video then generalizes these concepts to higher dimensions, where feasible regions become polyhedra. The approach provides an intuitive understanding of linear programming and its solutions.

What is the main benefit of understanding the geometrical interpretation of linear programs?

• It simplifies the mathematical calculations.

• It provides an intuitive understanding of linear programs and algorithms.

• It allows for the use of more complex algorithms.

• It eliminates the need for constraints.

In the given example, what is the objective function to be minimized?

• X + 3Y

• X - Y

• X + Y

• X - 2Y

Which inequality constraint ensures that feasible solutions are on the right of the y-axis?

• X - Y ≤ 2

• X + 3Y ≤ 3

• X ≥ 0

• Y ≤ X - 2

What shape do the feasible solutions form in the given linear program example?

• Circle

• Square

• Triangle

• Rectangle

How is the objective function X + Y represented in the geometrical interpretation?

• As a vertical line

• As a single point

• As a series of level sets

• As a horizontal line

What is the direction of the gradient descent for the objective function X + Y?

• Top left

• Top right

• Bottom right

• Bottom left

What is the gradient of the objective function X + Y?

• (1, 1)

• (1, 0)

• (-1, -1)

• (0, 1)

Where is the minimizer of the linear program located in the feasible region?

• Top right corner

• Bottom right corner

• Top left corner

• Bottom left corner

What is the minimizer point of the linear program in the example?

• (-3, 0)

• (3, 0)

• (0, -3)

• (0, 0)

What happens to the set of feasible solutions when there are more equality constraints?

• It becomes a line.

• It becomes a point.

• It remains a polygon.

• It becomes a circle.