Question 1

While the optimal solution to a constrained nonlinear model need not occur at an extreme point, it must occur at a boundary point.

Accepted Answer

In constrained nonlinear optimization, the optimal solution can occur at boundary points, interior points, or even along a curve or surface within the feasible region, depending on the nature of the objective function and constraints.

Question 2

One solution approach for solving a dynamic programming model is a backwards recursion approach.

Accepted Answer

In a backwards recursion approach, one starts at the final time period and works backwards to find the optimal solution for each time period.

Question 3

Dynamic programming is so named because:

Accepted Answer

Dynamic programming involves breaking down complex problems into smaller subproblems and solving them in a recursive manner. These subproblems usually have some sort of interdependence and are part of a larger multistage scenario. Therefore, option B is the most appropriate answer. While the optimal solutions may change over time, it is not the defining characteristic of dynamic programming, so option A is incorrect. Dynamic programming may involve parameters that vary, but this is not the fundamental feature of the approach, so option C is incorrect. Finally, while dynamic programming may involve decisions, this is not its unique feature, so option D is also incorrect.

Question 4

In a convex programming problem, while the objective function can have any shape, the set of constraints must form a convex set.

Accepted Answer

In a convex programming problem, both the objective function must be convex (or concave depending on the optimization direction) and the set of constraints must form a convex set to ensure the problem is convex.

Question 5

Which of the following may be said to have no single form?

Accepted Answer

Dynamic programming has no single form as it involves breaking down of complex problems into smaller problems and solving them one by one. There are different ways to approach and solve dynamic programming problems. Standard linear programming, integer linear programming, and binary linear programming have established forms and methods of solving them.

Question 6

Simply put, Bellman's principle of optimality states that the
optimal path to the end of the process does not depend upon how the
current state was reached.

Accepted Answer

Bellman's principle of optimality holds that the optimal solution for a problem can always be break down into optimal solutions to its subproblems. As a result, the optimal path to the end of the process does not depend upon how the current state was reached.

Question 7

Efficient solution procedures, guaranteed to provide optimal solutions, exist for convex programming and quadratic programming.

Accepted Answer

Efficient algorithms such as interior point methods and active-set methods exist for solving convex programming and quadratic programming problems to optimality.

Question 8

In a dynamic program, the boundary conditions refer to the first stage, and the stopping rule refers to the last stage.

Accepted Answer

The boundary conditions refer to the initial stage and the decision variables at that point, while the stopping rule determines when the optimization process should stop.

Question 9

LaGrange multipliers are like shadow prices in that they give the "instantaneous" change to the objective function value for changes to right hand side coefficients.

Accepted Answer

LaGrange multipliers are indeed similar to shadow prices in that they give the immediate impact on the objective function when there are changes to the constraints (right-hand side coefficients).

Question 10

Variables raised to a power other than 1, may be found in a nonlinear programming problem.Other nonlinearities include:

Accepted Answer

Both cross products and quotients can create nonlinearities in a programming problem. Therefore, option C is the correct choice. Option A and D are incorrect since one of them excludes quotients and the other excludes cross products, both of which can be nonlinear. Option B is also incorrect since quotients can create nonlinearities.

Question 11

According to goal programming proponents, most business problems have conflicting objectives and cannot be solved by optimizing a linear programming model with a single objective function.

Accepted Answer

The answer of According to goal programming proponents, most business...

Question 12

Sootaway Chimney Cleaners has a preemptive goal programming model for their three goals: reduce cost, reduce personnel, and raise quality.If goal 2 has a higher priority than goal 3, it is not possible for goal 3 to be met unless goal 2 has been met first.

Accepted Answer

In goal programming, priorities indicate the order in which goals are to be pursued, but achieving a lower priority goal is still possible without fully meeting a higher priority goal, as the model seeks to minimize the deviation from all set goals simultaneously, not sequentially.

Question 13

The optimal solution for an unconstrained concave function occurs where the slope equals zero for all variables.

Accepted Answer

This is true for unconstrained concave functions, which have a single global maximum. At the maximum, the slope of the function (its derivative) is zero in all directions.

Question 14

A goal programming problem can be transformed into a series of linear programming problems, each of which has a different objective function and one more constraint than the previous one in the series.

Accepted Answer

This statement is true. Goal programming involves formulating multiple conflicting goals or objectives, and transforming the problem into a series of linear programming problems with a single objective function for each goal and an increasing number of constraints. Each successive problem in the series adds a constraint that represents the deviation from the ideal goal. The aim is to find a solution that minimizes the deviations from all the goals while satisfying all the constraints.

Question 15

In applying Bellman's principle of optimality for dynamic programming using backwards recursion for a maximization problem, at a given state within a given stage:

Accepted Answer

Bellman's principle of optimality states that the optimal policy has the property that, whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. Thus, it does not matter how the state is reached.

Question 16

Altering the order of the stages of a dynamic program will produce a different result.

Accepted Answer

Dynamic programming relies on solving subproblems and combining their solutions in a specific order to solve a larger problem. The order of stages is determined by the problem's structure and dependencies between subproblems, not by arbitrary sequencing, ensuring consistent results regardless of the processing sequence as long as dependencies are respected.

Question 17

Quadratic programming is a special case of __________ programming.

Accepted Answer

Quadratic programming involves optimizing a quadratic objective function subject to linear constraints, which falls under the category of general nonlinear programming. Standard linear programming involves optimizing a linear objective function subject to linear constraints, while goal programming involves solving for multiple objectives simultaneously. Dynamic programming involves breaking down a problem into subproblems and solving each one individually, taking into account past decisions and future consequences.

Question 18

In a mathematical model with two variables, a function which is both concave and convex is a straight line.

Accepted Answer

A function that is both concave and convex is a function that has no local maxima or minima. The only type of function that satisfies this condition is a straight line.

Question 19

A knapsack problem can be modeled as a longest path network problem.

Accepted Answer

A knapsack problem can be modeled as a longest path network problem by creating a network with a source node, a sink node, and a node for each item in the knapsack. The edges between the nodes represent the possible combinations of items that can be placed in the knapsack, and the weights of the edges represent the total value of the items in each combination. The longest path from the source node to the sink node represents the optimal solution to the knapsack problem.

Question 20

For a convex nonlinear programming problem, the Kuhn-Tucker conditions are both necessary and sufficient.

Accepted Answer

The Kuhn-Tucker conditions are both necessary and sufficient for solving convex nonlinear programming problems.

Quiz 13: Nonlinear Models:dynamic, Goal, and Nonlinear Programming