Michael H. Veatch - Linear and Convex Optimization

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Linear and Convex Optimization: краткое содержание, описание и аннотация

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Discover the practical impacts of current methods of optimization with this approachable, one-stop resource Linear and Convex Optimization: A Mathematical Approach Experienced researcher and undergraduate teacher Mike Veatch presents the main algorithms used in linear, integer, and convex optimization in a mathematical style with an emphasis on what makes a class of problems practically solvable and developing insight into algorithms geometrically. Principles of algorithm design and the speed of algorithms are discussed in detail, requiring no background in algorithms.
The book offers a breadth of recent applications to demonstrate the many areas in which optimization is successfully and frequently used, while the process of formulating optimization problems is addressed throughout. 
Linear and Convex Optimization Coverage of current methods in optimization in a style and level that remains appealing and accessible for mathematically trained undergraduates Enhanced insights into a few algorithms, instead of presenting many algorithms in cursory fashion An emphasis on the formulation of large, data-driven optimization problems Inclusion of linear, integer, and convex optimization, covering many practically solvable problems using algorithms that share many of the same concepts Presentation of a broad range of applications to fields like online marketing, disaster response, humanitarian development, public sector planning, health delivery, manufacturing, and supply chain management Ideal for upper level undergraduate mathematics majors with an interest in practical applications of mathematics, this book will also appeal to business, economics, computer science, and operations research majors with at least two years of mathematics training.

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In this expression 75 is the weight per pallet of tents so is the weight of - фото 32

In this expression, 7.5 is the weight per pallet of tents, so картинка 33is the weight of tents. Similarly, картинка 34is the weight of food. The left‐hand side, then, is the total weight of the load, which must be less than or equal to the payload capacity of 40 (these quantities are in 1000s of lbs). The space limit requires that

Linear and Convex Optimization - изображение 35

The left‐hand side is the total number of pallets. This total does not have to be an integer; a total of 5.4 pallets would mean that one pallet is only loaded 40% full. Finally, only five pallets of food are ready, so

These inequalities define the domain of We will call them constraints and the - фото 36

These inequalities define the domain of картинка 37. We will call them constraints and the function картинка 38to be maximized the objective function . Optimizing a function whose domain is defined by constraints is a constrained optimization problem. The complete problem is

(1.1) We have abbreviated subject to as st Constrained optimization problems - фото 39

We have abbreviated “subject to” as “s.t.”

Constrained optimization problems have three components:

Components of an Optimization Problem

1 1. The decision variables.

2 2. The objective function to be maximized or minimized, as a function of the decision variables.

3 3. The constraints that the decision variables are subject to.

The formulation of this problem consists of the Eqs. ( 1.1) and the definitions of the variables. It is essential to define the variables and also good practice to label or describe the objective function and each constraint. This problem is an example of a linear program because the variables are continuous and the objective function and all constraints are linear.

Now that we have formulated the loading decisions as a linear program, how do we solve it? For картинка 40and картинка 41to satisfy the constraints, they must lie in the intersection of the half planes defined by these inequalities, shown in Figure 1.1. Most linear inequalities can be conveniently graphed by finding the картинка 42and картинка 43intercept of the corresponding equation, drawing a line between them, and checking a point not on the line, such as картинка 44, to see if it satisfies the inequality. If the point satisfies the inequality, then the half‐plane is on the same side of the line as that point; if the point does not satisfy the inequality, the half‐plane is on the other side of the line as that point. For the first constraint (weight), the картинка 45intercept is картинка 46, the картинка 47intercept is 8, and (0,0) is on the correct side of the line. Other constraints, such as картинка 48, have horizontal or vertical boundary lines. Once all of the constraints have been graphed, we can identify the region (or possibly a line or a point) satisfying all the constraints. We are seeking the point in this region that has the largest value of Linear and Convex Optimization - изображение 49. One way to find this point graphically is to plot contour lines Linear and Convex Optimization - изображение 50for one or two values of For example in Figure 11the contours Figure 11Region satisfying - фото 51. For example, in Figure 1.1the contours

Linear and Convex Optimization - изображение 52

Figure 1.1Region satisfying constraints for sending aid.

Linear and Convex Optimization - изображение 53

are graphed as dashed lines. They both intersect the shaded region, so there are points satisfying the constraints with objective function values of 24 and 36. Since all contour lines are parallel, we can visualize sweeping the line up and to the right without rotating it to find larger objective function values. The farthest contour line that touches the shaded region is shown in Figure 1.2. The point where it touches the region has the largest objective function value. This point lies on the constraint lines for weight and pallets, so we can find it by solving these two constraints with equality in place of inequality:

Linear and Convex Optimization - изображение 54

with solution Linear and Convex Optimization - изображение 55and objective function value 44. This agrees with Figure 1.2, where the contour line drawn is Linear and Convex Optimization - изображение 56. Thus, the optimal load is four pallets of tents and two pallets of food, with an expected value of 44.

Figure 12Optimal point and contour for sending aid To summarize to solve a - фото 57

Figure 1.2Optimal point and contour for sending aid.

To summarize, to solve a linear program with two variables graphically:

Solving a Linear Program Graphically

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