Nonlinear Programming

Question 1

The concept of __________ is now well rooted as a principle underlying the analysis of many complex decision or allocation problems. It offers a certain degree of philosophical elegance that is hard to dispute, and it often offers an indispensable
degree of operational simplicity.

Answer 1

Optimization

Question 2

Using this _________, one approaches a complex decision problem, involving the selection of values for a number of interrelated variables, by focusing attention on a single objective designed to quantify
performance and measure the quality of the decision. This one objective is maximized
(or minimized, depending on the formulation) subject to the constraints that may limit
the selection of decision variable values.

Answer 2

Optimization philosophy

Question 3

Optimization problems are usually divided into two major categories:

Answer 3

Linear and Nonlinear Programming

Question 4

These categories are distinguished by the presence or
absence of nonlinear functions in either the objective function or constraints and lead to very distinct solution methods.

Answer 4

Linear and nonlinear programming

Question 5

_________ programming assumes linearity. It is a technique that assumes that the relationships between the decision variables, the constraints, and the objective
function are linear, meaning that they can be represented by straight lines or planes.

Answer 5

Linear

Question 6

_______ programming embraces
the complexities of reality. It is a mathematical technique for
determining the optimal solution to many business problems.

Answer 6

Nonlinear

Question 7

LP problems can be solved using various algorithms, such as the

Answer 7

Simplex method, inter-point method, or branch-and-bound method

Question 8

It can also be used to model
problems such as resource allocation, production planning, transportation, scheduling,
and network flow.

Answer 8

Linear Programming

Question 9

It is more accurate compared to linear programs where it can be applied for the nonlinear
objective functions and constraints. The ____ techniques are based on a reduced gradient method utilizing the Lagrange multiplier or use the penalty function optimization approach.

Answer 9

Nonlinear Programming (NLP)

Question 10

In this method, the_________ (the
reduced gradient) is used to choose a search direction in the iterative procedure.

Answer 10

first partial derivatives of the equation

Question 11

_______ programming is a technique that allows for the relationships between
the decision variables, the constraints, and the objective function to be nonlinear,
meaning that they can be represented by curves or surfaces.

Answer 11

Nonlinear

Question 12

NLP problems can be more complex and challenging to solve than LP problems, and they may require different algorithms, such as the

Answer 12

gradient method, newton method, penalty method

Question 13

_________ can be used to model problems such as portfolio optimization, machine learning, engineering design, and economics.

Answer 13

Nonlinear Programming

Question 14

Slightly straightforward and fixed. Optimal solutions are to be found at the limited corner points of the convex polyhedron.

Answer 14

Linear Programming

Question 15

Varies. Optimal solutions can be anywhere along the boundaries of the feasible region or anywhere within the feasible region.

Answer 15

Nonlinear Programming

Question 16

Very easily amenable to linear algebraic transformation. Objective function have degree one.

Answer 16

Linear Programming

Question 17

Should be dealt with extreme care resulting in complex situations. Objective functions have degree more than one.

Answer 17

Nonlinear Programming

Question 18

Optimum always attained at constraint boundaries. A local optimum is also a global optimum

Answer 18

Linear programming

Question 19

Optima may be in the interior as well as at boundaries of constraints. A local optimum is not necessarily a global optimum

Answer 19

Nonlinear programming

Question 20

If either f (x1, x2, ……, xn) or some g1 (x1, x2, …… xn) or both are non-linear, then the problem of determining the n-type (x1, x2, …. xn) which makes z a minimum
or maximum and satisfies both (b) and (c), above is called a

Answer 20

General nonlinear programming problem (GNLPP)

Question 21

This is one of the reasons why all the non-linear programming problems cannot be grouped under the same title.

Answer 21

There is no known algorithm to effectively and efficiently solve a given general nonlinear programming problem. A method that is found to be useful in one problem may not be useful in another.

Question 22

Solving nonlinear equation depends on

Answer 22

● Number of decision variables
● Concavity of the function
● Present of constraints
● Complexity of objective function

Question 23

Steps In Identifying Method To Be Use

Answer 23

1. Define variables
2. Define requirements (constrained/unconstrained)
3. Form the objective function

Question 24

Nonlinear Programming Problem Classification

Answer 24

Unconstrained and Constrained

Question 25

Unconstrained

Answer 25

● One Dimensional Search
● Quadratic Forms
○ Derivatives
○ Taylor Series
○ Conditions for maximum/minimum
● Gradient Search
● Newton’s Method
● Quasi Newton Method

Question 26

Constrained

Answer 26

● Equality
○ Lagrange Multiplier Method
● Inequality
○ Karush-Kuhn Tucker
● Penalty Method

Question 27

It is a powerful technique used in optimization problems, in which the objective is to maximize or minimize a function while adhering to one or more constraints.

Answer 27

Lagrange Multiplier Method

Question 28

He is an Italian mathematician who introduced Lagrange Multiplier Method in 1804.

Answer 28

Joseph-Louis Lagrange

Question 29

The fundamental idea behind Lagrange Multiplier Method is to

Answer 29

convert a limited problem into a from that allows the derivative test of an unconstrained problem to be applied

Question 30

This method is particularly functional for constrained optimization situations.

Answer 30

Lagrange Multiplier Method

Question 31

It is used for determining the local maxima and minima of a function of the form f(x, y, z) under equality constraints of this type g(x, y, z) = k or g(x, y, z) =0.

Answer 31

Lagrange Multiplier Method

Question 32

It is denoted by L, it is a function created by combining the objective function with the constraints. An example of a mathematical representation of a ______ is:

L (x, y, λ) = f (x, y) + λg (x, y)

Answer 32

Lagrangian Function

Question 33

It indicates the number intended to maximize or minimize, such as _________ or any other measure of interest. In mathematical notation, the objective function is frequently denoted as:

f (x1, x2, x3…., xn), where x1, x2, x3…., xn are the decision variables.

Answer 33

profit, cost, utility

Question 34

Refers to the limitations or requirements that must be met in the process of optimization task. These ________ limit the available metrics for the decision
variables and frequently occur owing to real-world constraints, physical limitations, or
regulatory obligations.

Answer 34

Constraints

Question 35

Lagrange multipliers is denoted by λ, called ________. They are integers that indicate the “cost” of breaking each constraint. They are used to alter the objective function to account for constraints, allowing the optimization issue to be addressed effectively.

Answer 35

Lambda

Question 36

These are the points, where the Lagrangian function’s derivative, or gradient, is zero. These are potential solutions to the optimization problem and essential for identifying viable solutions that optimize the objective function while also satisfying the limitations.

Answer 36

Critical Points

Question 37

CASE 1: ONE CONSTRAINT AND TWO VARIABLES (Steps)

Answer 37

Step 1: Find the Lagrange’s Function
Step 2: Find stationary ‘z0’ (a,b) and Lagrange Multiplier ‘λ’ by using
Step 3: Find the Determinant ‘∆’
Step 4: Checking of maxima or minima

Question 38

CASE 2: ONE CONSTRAINT AND THREE VARIABLES (Steps)

Answer 38

Step 1: Find Lagrange’s Function
Step 2: Find stationary point ′z0′ (a,b,c) and ‘λ’ by using…
Step 3: Note: The computation for delta is another lesson…
Step 4: Since ∆ is negative
∴ Z0 will give minimum value at objective function
Step 5: Put X1, x2, x3 in Equation 4, Put λ = −2 in Equations 1, 2, & 3

Question 39

CASE 3: TWO CONSTRAINTS AND TWO VARIABLE (Steps)

Answer 39

Step 1: Find the Lagrange’s function
Step 2: Find stationary point Z0 (a, b) and λ1, λ2 by using
Step 3: Find border Hessian matrix and ∇4
Step 4: Determine the rules in identifying maxima or minima
Step 5: Substitute

Question 40

CASE 4: TWO CONSTRAINTS AND THREE VARIABLE (Steps)

Answer 40

Step 1: Find the Lagrange’s function
Step 2: Find stationary point Z0 (a, b) and λ1, λ2 by using
Step 3: Find border Hessian matrix and ∇5

Question 41

The optimality conditions for a constrained local optimum are called the

Answer 41

Karush Kuhn Tucker (KKT) conditions, also called as Kuhn-Tucker Conditions

Question 42

Kuhn-Tucker Conditions was originally named after: ______ & ________, who first published the
conditions in 1951; and ________, who stated the necessary conditions in his master’s thesis in 1939.

Answer 42

Harold W. Kuhn and Albert W. Tucker and William Karush

Question 43

Are the first derivative tests or first-order necessary conditions for a solution in nonlinear programming to be optimal in a mathematical optimization problem, provided that the set of regularity conditions are satisfied. They play an important role in constrained optimization theory and algorithm development.

Answer 43

KKT Conditions

Question 44

The Kuhn-Tucker necessary conditions for the problem Maximize Z = f(x) subject to the constraints gi(x) ≤ 0, i = 1, 2, … , m are also sufficient conditions if

Answer 44

f(x) is concave and gi(x) are convex functions of x

Question 45

CONDITIONS FOR:

Answer 45

A. One inequality constraint and two variables
B. Two inequality constraints and two variables