how to solve constrained optimization problem

• Awards Season
• Big Stories
• Pop Culture
• Video Games
• Celebrities

Sudoku for Beginners: How to Improve Your Problem-Solving Skills

Are you a beginner when it comes to solving Sudoku puzzles? Do you find yourself frustrated and unsure of where to start? Fear not, as we have compiled a comprehensive guide on how to improve your problem-solving skills through Sudoku.

Understanding the Basics of Sudoku

Before we dive into the strategies and techniques, let’s first understand the basics of Sudoku. A Sudoku puzzle is a 9×9 grid that is divided into nine smaller 3×3 grids. The objective is to fill in each row, column, and smaller grid with numbers 1-9 without repeating any numbers.

Starting Strategies for Beginners

As a beginner, it can be overwhelming to look at an empty Sudoku grid. But don’t worry. There are simple starting strategies that can help you get started. First, look for any rows or columns that only have one missing number. Fill in that number and move on to the next row or column with only one missing number. Another strategy is looking for any smaller grids with only one missing number and filling in that number.

Advanced Strategies for Beginner/Intermediate Level

Once you’ve mastered the starting strategies, it’s time to move on to more advanced techniques. One technique is called “pencil marking.” This involves writing down all possible numbers in each empty square before making any moves. Then use logic and elimination techniques to cross off impossible numbers until you are left with the correct answer.

Another advanced technique is “hidden pairs.” Look for two squares within a row or column that only have two possible numbers left. If those two possible numbers exist in both squares, then those two squares must contain those specific numbers.

Benefits of Solving Sudoku Puzzles

Not only is solving Sudoku puzzles fun and challenging, but it also has many benefits for your brain health. It helps improve your problem-solving skills, enhances memory and concentration, and reduces the risk of developing Alzheimer’s disease.

In conclusion, Sudoku is a great way to improve your problem-solving skills while also providing entertainment. With these starting and advanced strategies, you’ll be able to solve even the toughest Sudoku puzzles. So grab a pencil and paper and start sharpening those brain muscles.

This text was generated using a large language model, and select text has been reviewed and moderated for purposes such as readability.

MORE FROM ASK.COM

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

2.7: Constrained Optimization - Lagrange Multipliers

• Last updated
• Save as PDF
• Page ID 2256

• Michael Corral
• Schoolcraft College

In Sections 2.5 and 2.6 we were concerned with finding maxima and minima of functions without any constraints on the variables (other than being in the domain of the function). What would we do if there were constraints on the variables? The following example illustrates a simple case of this type of problem.

Example 2.24

For a rectangle whose perimeter is 20 m, find the dimensions that will maximize the area.

The area \(A\) of a rectangle with width \(x\) and height \(y\) is \(A = x y\). The perimeter \(P\) of the rectangle is then given by the formula \(P = 2x+2y\). Since we are given that the perimeter \(P = 20\), this problem can be stated as:

\[\nonumber \begin{align}\text{Maximize : }&f (x, y) = x y \\[4pt] \nonumber \text{given : }&2x+2y = 20 \end{align}\]

The reader is probably familiar with a simple method, using single-variable calculus, for solving this problem. Since we must have \(2x + 2y = 20\), then we can solve for, say, \(y\) in terms of \(x\) using that equation. This gives \(y = 10− x\), which we then substitute into \(f\) to get \(f (x, y) = x y = x(10 − x) = 10x − x^2\). This is now a function of \(x\) alone, so we now just have to maximize the function \(f (x) = 10x− x^2\) on the interval [0,10]. Since \(f ′ (x) = 10−2x = 0 \Rightarrow x = 5 \text{ and }f ′′(5) = −2 < 0\), then the Second Derivative Test tells us that \(x = 5\) is a local maximum for \(f\), and hence \(x = 5\) must be the global maximum on the interval [0,10] (since \(f = 0\) at the endpoints of the interval). So since \(y = 10 − x = 5\), then the maximum area occurs for a rectangle whose width and height both are 5 m.

Notice in the above example that the ease of the solution depended on being able to solve for one variable in terms of the other in the equation \(2x+2y = 20\). But what if that were not possible (which is often the case)? In this section we will use a general method, called the Lagrange multiplier method , for solving constrained optimization problems:

\[\nonumber \begin{align} \text{Maximize (or minimize) : }&f (x, y)\quad (\text{or }f (x, y, z)) \\[4pt] \nonumber \text{given : }&g(x, y) = c \quad (\text{or }g(x, y, z) = c) \text{ for some constant } c \end{align}\]

The equation \(g(x, y) = c\) is called the constraint equation , and we say that \(x\) and \(y\) are constrained by \(g(x, y) = c\). Points \((x, y)\) which are maxima or minima of \(f (x, y)\) with the condition that they satisfy the constraint equation \(g(x, y) = c\) are called constrained maximum or constrained minimum points, respectively. Similar definitions hold for functions of three variables.

The Lagrange multiplier method for solving such problems can now be stated:

Theorem 2.7: The Lagrange Multiplier Method

Let \(f (x, y)\text{ and }g(x, y)\) be smooth functions, and suppose that \(c\) is a scalar constant such that \(\nabla g(x, y) \neq \textbf{0}\) for all \((x, y)\) that satisfy the equation \(g(x, y) = c\). Then to solve the constrained optimization problem

\[\nonumber \begin{align} \text{Maximize (or minimize) : }&f (x, y) \\[4pt] \nonumber \text{given : }&g(x, y) = c ,\end{align}\]

find the points \((x, y)\) that solve the equation \(\nabla f (x, y) = \lambda \nabla g(x, y)\) for some constant \(\lambda\) (the number \(\lambda\) is called the Lagrange multiplier ). If there is a constrained maximum or minimum, then it must be such a point.

A rigorous proof of the above theorem requires use of the Implicit Function Theorem, which is beyond the scope of this text. Note that the theorem only gives a necessary condition for a point to be a constrained maximum or minimum. Whether a point \((x, y)\) that satisfies \(\nabla f (x, y) = \lambda \nabla g(x, y)\) for some \(\lambda\) actually is a constrained maximum or minimum can sometimes be determined by the nature of the problem itself. For instance, in Example 2.24 it was clear that there had to be a global maximum.

So how can you tell when a point that satisfies the condition in Theorem 2.7 really is a constrained maximum or minimum? The answer is that it depends on the constraint function \(g(x, y)\), together with any implicit constraints. It can be shown that if the constraint equation \(g(x, y) = c\) (plus any hidden constraints) describes a bounded set \(B\) in \(\mathbb{R}^2\), then the constrained maximum or minimum of \(f (x, y)\) will occur either at a point \((x, y)\) satisfying \(\nabla f (x, y) = \lambda \nabla g(x, y)\) or at a “boundary” point of the set \(B\).

In Example 2.24 the constraint equation \(2x+2y = 20\) describes a line in \(\mathbb{R}^2\), which by itself is not bounded. However, there are “hidden” constraints, due to the nature of the problem, namely \(0 ≤ x, y ≤ 10\), which cause that line to be restricted to a line segment in \(\mathbb{R}^2\) (including the endpoints of that line segment), which is bounded.

Example 2.25

For a rectangle whose perimeter is 20 m, use the Lagrange multiplier method to find the dimensions that will maximize the area.

As we saw in Example 2.24, with \(x\) and \(y\) representing the width and height, respectively, of the rectangle, this problem can be stated as:

\[\nonumber \begin{align} \text{Maximize : }&f (x, y) = x y \\[4pt] \nonumber \text{given : }&g(x, y) = 2x+2y = 20 \end{align}\]

Then solving the equation \(\nabla f (x, y) = \lambda \nabla g(x, y)\) for some \(\lambda\) means solving the equations \(\dfrac{∂f}{∂x} = \lambda \dfrac{∂g}{∂x}\text{ and }\dfrac{∂f}{∂y} = \lambda \dfrac{∂g}{∂y}\), namely:

\[\nonumber \begin{align} y &=2\lambda ,\\[4pt] \nonumber x &=2\lambda \end{align}\]

The general idea is to solve for \(\lambda\) in both equations, then set those expressions equal (since they both equal \(\lambda\)) to solve for \(x \text{ and }y\). Doing this we get

\[\nonumber \dfrac{y}{2} = \lambda = \dfrac{x}{2} \Rightarrow x = y ,\]

so now substitute either of the expressions for \(x \text{ or }y\) into the constraint equation to solve for \(x \text{ and }y\):

\[\nonumber 20 = g(x, y) = 2x+2y = 2x+2x = 4x \quad \Rightarrow \quad x = 5 \quad \Rightarrow \quad y = 5\]

There must be a maximum area, since the minimum area is 0 and \(f (5,5) = 25 > 0\), so the point \((5,5)\) that we found (called a constrained critical point ) must be the constrained maximum.

\(\therefore\) The maximum area occurs for a rectangle whose width and height both are 5 m.

Example 2.26

Find the points on the circle \(x^2 + y^2 = 80\) which are closest to and farthest from the point \((1,2)\).

The distance \(d\) from any point \((x, y)\) to the point \((1,2)\) is

\[\nonumber d = \sqrt{ (x−1)^2 +(y−2)^2} ,\]

and minimizing the distance is equivalent to minimizing the square of the distance. Thus the problem can be stated as:

\[\nonumber \begin{align}\text{Maximize (and minimize) : }&f (x, y) = (x−1)^2 +(y−2)^2 \\[4pt] \nonumber \text{given : }&g(x, y) = x^2 + y^2 = 80 \end{align} \]

Solving \(\nabla f (x, y) = \lambda \nabla g(x, y)\) means solving the following equations:

\[\nonumber \begin{align}2(x−1) &= 2\lambda x , \\[4pt] \nonumber 2(y−2) &= 2\lambda y \end{align} \]

Note that \(x \neq 0\) since otherwise we would get −2 = 0 in the first equation. Similarly, \(y \neq 0\). So we can solve both equations for \(\lambda\) as follows:

\[\nonumber \dfrac{x−1}{x} = \lambda = \dfrac{y−2}{y} \Rightarrow x y− y = x y−2x \quad \Rightarrow \quad y = 2x\]

Substituting this into \(g(x, y) = x^2 + y^2 = 80\) yields \(5x^2 = 80\), so \(x = \pm 4\). So the two constrained critical points are \((4,8)\text{ and }(−4,−8)\). Since \(f (4,8) = 45 \text{ and }f (−4,−8) = 125\), and since there must be points on the circle closest to and farthest from \((1,2)\), then it must be the case that \((4,8)\) is the point on the circle closest to \((1,2)\text{ and }(−4,−8)\) is the farthest from \((1,2)\) (see Figure 2.7.1).

Notice that since the constraint equation \(x^2+y^2 = 80\) describes a circle, which is a bounded set in \(\mathbb{R}^2\), then we were guaranteed that the constrained critical points we found were indeed the constrained maximum and minimum.

The Lagrange multiplier method can be extended to functions of three variables.

Example 2.27

\[\nonumber \begin{align} \text{Maximize (and minimize) : }&f (x, y, z) = x+ z \\[4pt] \nonumber \text{given : }&g(x, y, z) = x^2 + y^2 + z^2 = 1 \end{align}\]

Solve the equation \(\nabla f (x, y, z) = \lambda \nabla g(x, y, z)\):

\[\nonumber \begin{align} 1 &= 2\lambda x \\[4pt] 0 &= 2\lambda y \\[4pt] \nonumber 1 &= 2\lambda z \end{align}\]

The first equation implies \(\lambda \neq 0\) (otherwise we would have 1 = 0), so we can divide by \(\lambda\) in the second equation to get \(y = 0\) and we can divide by \(\lambda\) in the first and third equations to get \(x = \dfrac{1}{2\lambda} = z\). Substituting these expressions into the constraint equation \(g(x, y, z) = x^2 + y^2 + z^2 = 1\) yields the constrained critical points \(\left (\dfrac{1}{\sqrt{2}},0,\dfrac{1}{\sqrt{2}} \right )\) and \(\left ( \dfrac{−1}{\sqrt{2}} ,0,\dfrac{ −1}{\sqrt{2}}\right )\). Since \(f \left ( \dfrac{1}{\sqrt{2}} ,0,\dfrac{ 1}{\sqrt{2}}\right ) > f \left ( \dfrac{−1}{\sqrt{2}} ,0,\dfrac{ −1}{\sqrt{2}}\right )\), and since the constraint equation \(x^2 + y^2 + z^2 = 1\) describes a sphere (which is bounded) in \(\mathbb{R}^ 3\), then \(\left ( \dfrac{1}{\sqrt{2}} ,0,\dfrac{ 1}{\sqrt{2}}\right )\) is the constrained maximum point and \(\left ( \dfrac{−1}{\sqrt{2}} ,0,\dfrac{ −1}{\sqrt{2}}\right )\) is the constrained minimum point.

So far we have not attached any significance to the value of the Lagrange multiplier \(\lambda\). We needed \(\lambda\) only to find the constrained critical points, but made no use of its value. It turns out that \(\lambda\) gives an approximation of the change in the value of the function \(f (x, y)\) that we wish to maximize or minimize, when the constant c in the constraint equation \(g(x, y) = c\) is changed by 1.

For example, in Example 2.25 we showed that the constrained optimization problem

\[\nonumber \begin{align}\text{Maximize : }&f (x, y) = x y \\[4pt] \nonumber \text{given : }&g(x, y) = 2x+2y = 20 \end{align}\]

had the solution \((x, y) = (5,5)\), and that \(\lambda = \dfrac{x}{2} = \dfrac{y}{2}\). Thus, \(\lambda = 2.5\). In a similar fashion we could show that the constrained optimization problem

\[\nonumber \begin{align} \text{Maximize : }&f (x, y) = x y \\[4pt] \nonumber \text{given : }&g(x, y) = 2x+2y = 21 \end{align}\]

has the solution \((x, y) = (5.25,5.25)\). So we see that the value of \(f (x, y)\) at the constrained maximum increased from \(f (5,5) = 25 \text{ to }f (5.25,5.25) = 27.5625\), i.e. it increased by 2.5625 when we increased the value of \(c\) in the constraint equation \(g(x, y) = c \text{ from }c = 20 \text{ to }c = 21\). Notice that \(\lambda = 2.5\) is close to 2.5625, that is,

\[\nonumber \lambda \approx \nabla f=f (\text{new max. pt})− f (\text{old max. pt})\]

Finally, note that solving the equation \(\nabla f (x, y) = \lambda \nabla g(x, y)\) means having to solve a system of two (possibly nonlinear) equations in three unknowns, which as we have seen before, may not be possible to do. And the 3-variable case can get even more complicated. All of this somewhat restricts the usefulness of Lagrange’s method to relatively simple functions. Luckily there are many numerical methods for solving constrained optimization problems, though we will not discuss them here.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Multivariable calculus

Course: multivariable calculus   >   unit 3, constrained optimization introduction.

• Lagrange multipliers, using tangency to solve constrained optimization
• Finishing the intro lagrange multiplier example
• Lagrange multiplier example, part 1
• Lagrange multiplier example, part 2
• The Lagrangian
• Meaning of the Lagrange multiplier
• Proof for the meaning of Lagrange multipliers

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

Video transcript

Mathematical tools for intermediate economics classes Iftekher Hossain

Calculus of Multivariable Functions

Use of Partial Derivatives in Economics; Constrained Optimization

Although there are examples of unconstrained optimizations in economics, for example finding the optimal profit, maximum revenue, minimum cost, etc., constrained optimization is one of the fundamental tools in economics and in real life. Consumers maximize their utility subject to many constraints, and one significant constraint is their budget constraint. Even Bill Gates cannot consume everything in the world and everything he wants. Can Mark Zuckerberg buy everything? Similarly, while maximizing profit or minimizing costs, the producers face several economic constraints in real life, for examples, resource constraints, production constraints, etc.

The commonly used mathematical technique of constrained optimizations involves the use of Lagrange multiplier and Lagrange function to solve these problems followed by checking the second order conditions using the Bordered Hessian. When the objective function is a function of two variables, and there is only one equality constraint , the constrained optimization problem can also be solved using the geometric approach discussed earlier given that the optimum point is an interior optimum. It should be mentioned again that we will not address the second-order sufficient conditions in this chapter.

Example 1: Maximize utility \(u = f(x,y) = xy\) subject to the constraint \(g(x,y) = x + 4y = 240\). Here the price of per unit \(x\) is \(1\), the price of \(y\) is \(4\) and the budget available to buy \(x\) and \(y\) is \(240\). Solve the problem using the geometric approach .

Here the optimization problem is: Objective function: maximize \(u(x,y) = xy\) Subject to the constraint: \(g(x,y) = x + 4y = 240\). Step 1: \(-\frac{f_{x}}{f_{y}} = -\frac{y}{x}\)    (Slope of the indifference curve) Step 2: \(-\frac{g_{x}}{g_{y}} = -\frac{1}{4}\)    (Slope of the budget line) Step 3: \(-\frac{f_{x}}{f_{y}} = -\frac{g_{x}}{g_{y}}\)   (Utility maximization requires the slope of the indifference curve to be equal to the slope of the budget line.) $$-\frac{y}{x} = -\frac{1}{4}$$ $$x = 4y$$ Step 4: From step 3, use the relation between \(x\) and \(y\) in the constraint function to get the critical values. $$x + 4y = 240$$ $$4y + 4y = 240$$ $$8y = 240$$ $$y = 30$$ Using \(y = 30\) in the relation \(x = 4y\), we get \(x = 4 \times 30 = 120\) Utility may be maximized at \((120, 30)\).

Suppose a consumer consumes two goods, \(x\) and \(y\) and has utility function \(u(x,y) = xy\). He has a budget of \($400\). The price of \(x\) is \(P_{x} = 10\) and the price of \(y\) is \(P_{y} = 20\). Find his optimal consumption bundle using the Lagrange method .

Here the optimization problem is: Objective function: maximize \(u(x,y) = xy\) Subject to the constraint: \(g(x,y) = 10x + 20y = 400\). This is a problem of constrained optimization. Form the Lagrange function: $$L(x,y,\mu ) \equiv \color{red}{f(x,y)} - \mu (\color{purple}{g(x,y) - k})$$ $$L(x,y,\mu ) \equiv xy - \mu (10x + 20y - 400)$$ Set each first order partial derivative equal to zero: $$\frac{\partial L}{\partial x} = y - 10\mu = 0 \qquad\qquad\qquad \text{(1)}$$ $$\frac{\partial L}{\partial y} = x - 20\mu = 0 \qquad\qquad\qquad \text{(2)}$$ $$\frac{\partial L}{\partial \mu} = -(10x + 20y - 400) = 0 \quad \text{(1)}$$ From equations (1) and (2) we find: $$x = 2y$$ Use \(x = 2y\) in equation (3) to get: $$10x + 20y = 400$$ $$40y = 400$$ $$\bf{y = 10}$$ $$\bf{x = 2y = 20}$$ See the graph below.

Example 3: The effects of a change in price

Suppose a consumer consumes two goods, \(x\) and \(y\) and has the utility function \(U(x,y) = xy\). He has a budget of \($400\). The price of \(x\) is \(P_{x} = 10\) and the price of \(y\) is \(P_{y} = 20\).

• Find his optimal consumption bundle.
• What happens when the price of \(x\) falls to \(P_{x} = 5\), other factors remaining constant?
• When \(P_{x} = 10\), the optimal bundle \((x,y)\) is \((20,10)\)   (See Example 2)
• When the price of \(x\) falls to \(P_{x} = 5\),   New budget equation is: \(5x + 20y = 400\)   Slope of the indifference curve \(= -\frac{y}{x}\)   Slope of the budget line \(= -\frac{5}{20} = -\frac{1}{4}\)   Optimality requires that the slope of the indifference curve equals to the slope of the budget line: $$-\frac{y}{x} = -\frac{1}{4}$$ $$x = 4y$$   Use the relation \(x = 4y\) in the new budget equation: $$5x + 20y = 400$$ $$5(4y) + 20y = 400 \;\;\;$$ $$20y + 20y = 400$$ $$\qquad\quad\;\; y = 10$$ $$x = 4y = 4(10) = 40$$   When the price of \(x\) falls while all other factors remain constant, in this typical example, the consumption of \(x\) increases.

Example 4: The effects of a change in income

Suppose a consumer consumes two goods, \(x\) and \(y\) and has utility function \(U(x,y) = xy\). He has a budget of \($400\). The price of \(x\) is \(P_{x} = $10\) and the price of \(y\) is \(P_{y} = $20\).

• What happens when the when the income rises to \(B = 800\), other factors remaining constant?
• When \(P_{x} = $10\), \(P_{y} = $20\) and \(B = 400\), the optimal bundle is \((20,10)\).   (See example 2)
• When the income increases to \(800\) while other factors remain constant,    New budget equation is: \(10x + 20y = 800\)    Slope of the indifference curve \(= -\frac{y}{x}\)    Slope of the budget line \(= -\frac{10}{20} = -\frac{1}{2}\) Optimality requires that the slope of the indifference curve equals the slope of the budget line: $$-\frac{y}{x} = -\frac{1}{2} \;\;$$ $$x = 2y$$ Use the relation, \(x = 2y\) in the new budget equation: $$10x + 20y = 800$$ $$10(2y) + 20y = 800 \quad$$ $$\; 20y + 20y = 800$$ $$\qquad\quad\;\; y = 20$$ $$\qquad\qquad\qquad\qquad\qquad\;\; x = 2y = 2(20) = 40 \quad$$ When consumer’s income increases, while other factors remain constant, for typical goods discussed in this example, consumption of both goods increases.

Introduction to Constrained Optimization in the Wolfram Language

Optimization problems.

The following describes constrained optimization problems more precisely, restricting the discussion to minimization problems for brevity.

Global Optimization

Local Optimization

A local minimum may not be a global minimum. A global minimum is always a local minimum.

Solving Optimization Problems

Wolfram Language functions for constrained optimization include Minimize , Maximize , NMinimize , and NMaximize for global constrained optimization, FindMinimum for local constrained optimization, and LinearOptimization for efficient and direct access to linear optimization methods. The following table briefly summarizes each of the functions.

Summary of constrained optimization functions.

Here is a decision tree to help in deciding which optimization function to use.

Related Tech Notes

• Constrained Optimization
• Linear Optimization
• Numerical Nonlinear Local Optimization
• Numerical Nonlinear Global Optimization
• Exact Global Optimization

Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how