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• Prof. Dimitris Bertsimas

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• Sloan School of Management

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Introduction to mathematical programming, course description.

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• Chapter 1: Mathematical Programming

Mathematical programming is a powerful tool with broad applicability across various sectors. In this series, we will delve into the origins, relevance, and practical applications of mathematical programming, with a particular emphasis on Linear Programming (LP) and Mixed Integer Linear Programming (MIP) models.

History of Mathematical Programming

Mathematical programming, also known as mathematical optimization, originated with the invention of linear programming by George Dantzig in 1947. Since then, it has become an indispensable tool for decision-making and resource allocation in a wide range of industries, including finance, logistics, manufacturing, and transportation.

The Key Components of Mathematical Programming

The field of mathematical programming encompasses a three-step process.

• Create your mathematical model . You start by translating your real-world problems mathematically, defining the questions you’re asking (decision variables), your limitations (constraints), and the goals you need to achieve (objectives).
• Develop algorithms . Develop algorithms that solve these mathematical programming models. Thankfully, there are many mathematical programming “solver” solutions available, including Gurobi, that include the algorithms you’ll need.
• Run the algorithms . Finally, you run your model through the solver to find the answer to your problem (i.e., answers to your questions, based on your unique objectives and constraints).

The Difference Between Mathematical Programming and Computer Programming

Mathematical programming is a problem-solving approach that uses mathematical models and algorithms to optimize decision-making processes. Computer programming, on the other hand, is about writing code to create software or systems that computers can execute. While they both involve the word “programming,” they have different focuses and objectives.

Exploring Linear Programming

Linear Programming (LP) is a widely used mathematical programming technique that involves optimizing (minimizing or maximizing) a linear objective function (your defined goals) subject to a set of linear constraints (your defined limitations). LP is particularly useful in situations where resources need to be allocated efficiently or where decisions need to be made to maximize or minimize a certain outcome.

To illustrate the concepts of LP, we will introduce a typical case study known as the “Furniture Problem.” Throughout this series, we will use the Furniture Problem to demonstrate the step-by-step process of formulating and solving LP models. By applying LP techniques to this practical scenario, you will gain a comprehensive understanding of how mathematical programming can be applied in real-world situations.

In addition to the case study, we will provide a general formulation for LP and MIP problems. Understanding the basic structure and components of LP models will enable you to tackle a wide range of optimization problems effectively.

Download the complete  Linear Programming Tutorial Series slide deck .

View the entire series:

• Welcome: Linear Programming Tutorial
• Chapter 2: Introduction to Linear Programming
• Chapter 3: Mixed Integer Linear Programming Problems
• Chapter 4: Furniture Factory Problem
• Chapter 5: Simplex Method
• Chapter 6: Modeling and Solving Linear Programming Problems
• Chapter 7: Sensitivity Analysis of Linear Programming Problems
• Chapter 8: Multiple Optimal Solutions
• Chapter 9: Unbounded Linear Programming Problems
• Chapter 10: Infeasible Linear Programming Problems
• Chapter 11: Linear Programming Overview – Further Considerations
• Chapter 12: Duality in Linear Programming
• Chapter 13: Optimality Conditions
• Chapter 14: Dual Simplex Method

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Mathematical programming

2020 Mathematics Subject Classification: Primary: 90Cxx [ MSN ][ ZBL ]

The branch of mathematics concerned with the theory and methods for solving problems on finding the extrema of functions on sets defined by linear and non-linear constraints (equalities and inequalities) in a finite-dimensional vector space . Mathematical programming is a branch of operations research , which comprises a wide class of control problems the mathematical models of which are finite-dimensional extremum problems. The problems of mathematical programming find applications in various areas of human activity where it is necessary to choose one of the possible ways of action, e.g. in solving numerous problems of control and planning of production processes as well as in problems of design and long-term planning. The term "mathematical programming" is connected with the fact that the goal of solving various problems is choosing programs of action.

The mathematical formulation of the problem of mathematical programming is: To minimize a scalar function $\phi(x)$ of a vector argument on the set \begin{equation} X=\left\{x\colon q_i(x)\geq 0,\; i=1,\ldots,k;\quad h_j(x)=0,\; j=1,\ldots,m \right\}, \end{equation} where $q_i(x)$ and $h_j(x)$ are scalar functions. The function $\phi(x)$ is called the objective function, and also the quality criterion, the set $X$ is called the feasible set, or the set of plans, a solution $x^*$ of the mathematical programming problem is an optimum point (or vector), a point of global minimum and also an optimal plan.

In mathematical programming one customarily distinguishes the following branches.

• Linear programming : The objective function $\phi(x)$ and the constraints $q_i(x)$ and $h_j(x)$ are linear.
• Quadratic programming : The objective function is quadratic and convex, while the feasible set is defined by linear equalities and inequalities.
• Convex programming : The objective function and the constraints are convex.
• Discrete programming / Integer programming : The solution is sought only at discrete, say integer, points of $X$.
• Stochastic programming : In contrast to deterministic problems, here the data contain an element of indeterminacy. For example, in stochastic problems of minimization of a linear function $\sum_{j=1}^{n}c_j x_j$ under linear constraints $\sum_{j=1}^{n}a_{ij} x_j\geq b_i$, $i=1,2,\ldots$, the parameters $c_j$, $a_{ij}$, $b_i$, or only some of them, are random.

The problems of linear, quadratic and convex programming have a common property: Local optimality implies global optimality. The so-called multi-extremum problems , for which the indicated property does not hold, are both considerably more difficult and less investigated.

At the basis of the theory of convex, and, in particular, linear and quadratic, programming lie the Karush-Kuhn-Tucker conditions , which gives necessary and sufficient conditions for the existence of an optimum point $x^*$: In order that $x^*$ be an optimum point, i.e. \begin{equation} \phi(x^*)=\min_{x\in X}\phi(x), \end{equation} \begin{equation} X=\left\{x\colon f_i(x)\geq 0,\; i=1,\ldots,k\right\}, \end{equation} it is necessary and sufficient that there exist a point $y^*=(y_1^*,\ldots,y_k^*)$ such that the pair $x^*$, $y^*$ be a saddle point of the Lagrange function \begin{equation} L(x,y)=\phi(x)+\sum_{i=1}^{k}y_i f_i(x) . \end{equation} The latter means that \begin{equation} L(x^*,y)\leq L(x^*,y^*)\leq L(x,y^*), \end{equation} for all $x$ and all $y\geq 0$. If the constraints $f_i(x)$ are non-linear, the theorem is valid under some additional assumptions on the feasible set, for example, the existence of a point $x\in X$ such that $f_i(x)>0$, $i=1,\ldots,k$ (Slater's regularity condition).

If $\phi(x)$ and $f_i(x)$ are differentiable functions , then the following relations characterize a saddle point:

Thus, the problem of convex programming reduces to the solution of a system of equations and inequalities. \begin{equation} \begin{aligned} \sum_{j=1}^{n}\frac{\partial L}{\partial x_j}x_j = 0;&\qquad \frac{\partial L}{\partial x_j} \geq 0,& j=1,\ldots,n;\\ \sum_{i=1}^{k}\frac{\partial L}{\partial y_i}y_i = 0;&\qquad \frac{\partial L}{\partial y_i} \leq 0,\quad y_i\geq 0,& i=1,\ldots,k. \end{aligned} \end{equation} On the basis of the Karush-Kuhn-Tucker conditions various iterative minimization methods were developed, which amount to searching for a saddle point of Lagrange's function.

In mathematical programming one of the main directions concerns computational methods for solving extremum problems. One of the widest used among these methods is the method of feasible directions. In this method a sequence $\{x_m\}$ of points of the set $X$ is constructed by the formula $x_{p+1}=x_p + \alpha_p s_p$. At each iteration, to calculate the point $x_{p+1}$ one has to choose a direction (a vector) $s_p$ and a step length (a number) $\alpha_p$. For $s_p$ one selects from among the feasible directions (i.e. directions at the point $x_p$, a small displacement along which does not lead out of the set $X$) the one that makes an acute angle with the direction $g(x_p)$ of steepest descent of the objective function (with the vector $g(x_p)=-\left.\frac{d \phi(x)}{dx}\right|_{x=x_p}$ if $\phi(x)$ is differentiable). Therefore, along the direction $s_p$ the function $\psi_p(\alpha)=\phi(x_p + \alpha s_p)$ is decreasing. The number $\alpha_p$ is determined from the conditions that $x_{p+1}\in X$ and $\phi(x_{p+1})<\phi(x_p)$. To calculate $s_p$ at $x_p$ one defines the cone of feasible directions, given by a system of inequalities, and one formulates the problem (of linear programming) of finding the possible direction along which the objective function decreases most quickly. This problem is readily solved using, say, the standard simplex method . The step length $\alpha_p$ is determined by solving the minimization problem for the function $\psi_p(\alpha)$. Under rather general assumptions it has been shown that the distance from $x_p$ to the set of stationary points of the original problem (i.e. the set of points at which the conditions necessary for a local minimum of $\phi(x)$ on $X$ are satisfied) tends to zero as $p\rightarrow\infty$. In the case when the original problem is a problem in convex programming, then, as $p\rightarrow\infty$, $x_p$ approaches the set of solutions (optimum points) of the original problem. The method of feasible directions admits an approximate solution of the indicated problems of determination of $s_p$ and $\alpha_p$, and in this sense it is stable with respect to computational errors.

For feasible sets of special structure (from the point of view of the simplicity of choosing $s_p$) one can apply minimization methods different from the method of feasible directions. Thus, in a gradient-projection method, as the descent direction at the point $x_p$ one takes $s_p=y_p - x_p$, where $y_p$ is the projection of $v_p=x_p+g(x_p)$, $g(x_p)=-\left.\frac{d \phi(x)}{dx}\right|_{x=x_p}$, on the set $X$.

If $X$ is given by linear constraints, one has the rather promising method of the conditional gradient: One linearizes the objective function at $x_p$ by constructing the function $l(x)=\phi(x)+\langle g(x_p),x_p - x\rangle$, and then, minimizing $l(x)$ on the set $X$, one finds a minimum point $y_p$ of it and one finally sets $s_p=y_p - x_p$.

Minimization methods for non-smooth functions were successfully elaborated. One of the representatives of this class is the method of the generalized gradient. By definition, a generalized gradient at $x_p$ is any vector $\tilde{g}(x_p)$ such that the inequality $\phi(y)-\phi(x_p)\geq\langle\tilde{g}(x_p),y-x_p\rangle$ holds for all $y\in X$. This method differs from the projection-gradient method in that for the point that is projected one takes $v_p=x_p-\tilde{g}(x_p)$.

Stochastic methods of minimization are also widely used. In the simplest variant of the method of random descent one takes for $s_p$ a random point on the $n$-dimensional unit sphere with center at the coordinate origin, and if $s_p$ is a feasible direction and if, in addition, $\phi(x)$ decreases along this direction, then a step is effected, i.e. one shifts to the point $x_{p+1}$. In the opposite case, the procedure for selecting $s_p$ is carried out anew.

A characteristic feature of the computational aspect of the methods for solving the problems in mathematical programming is that the application of these methods is invariably connected with the utilization of electronic computers. The main reason for this is that the problems in mathematical programming that formalize situations of control of real systems involve a large amount of work which cannot be performed by manual computation.

One of the widespread methods for investigating problems in mathematical programming is the method of penalty functions . This method essentially replaces the given mathematical programming problem by a sequence of parametric problems on unconditional minimization, such that when the parameter tends to infinity (in other cases, to zero) the solutions of these auxiliary problems converge to the solution of the original problem. Note that in an unconditional minimization problem every direction is feasible, and consequently the search for $s_p$ requires less effort in comparison with the solution of the same problem by the method of feasible directions. (For example, in the method of steepest descent one takes $s_p=-\left.\frac{d \phi(x)}{dx}\right|_{x=x_p}$.) The same is true concerning the search for $\alpha_p$.

An important direction of investigation in mathematical programming is the problem of stability. Here essential importance is attached to the study of the class of stable problems, that is, problems for which small perturbations (errors) in the data result in perturbations of the solutions that are also small. In the case of unstable problems an important role is reserved for a procedure of approximating the unstable problem by a sequence of stable problems — this is known as the regularization process.

Along with finite-dimensional problems, problems of mathematical programming in infinite-dimensional spaces are considered as well. Among the latter there are various extremum problems in mathematical economics, technology, problems of optimization of physical characteristics of nuclear reactors, and others. In the terminology of mathematical programming one formulates the problems in the corresponding function space of variational calculus and optimal control .

Mathematical programming has crystallized as a science in the 1950-s until 1970-s. This is first of all connected with the development of electronic computers and, hence, the mathematical processing of large amounts of data. On this basis one solves problems of control and planning, in which the application of mathematical methods is connected mainly with the construction of mathematical models and related extremal problems, among them problems of mathematical programming.

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14. Solving Linear Programming Problems

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The following topics are included in this series of two videos.

• Solving the linear programming problem in video 1 in Chapter 13.
• Solving the linear programming problem in video 2 in Chapter 13.

• A florist makes 2 special bouquets. Both types consist of Japanese irises and tulips. Type I consists of 1 dozen tulips and 1 dozen Japanese irises. Type II consists of 2 dozen tulips and 4 dozen Japanese irises. The profit on each Type I bouquet is $8 and the profit on each Type II bouquet is $18. The florist knows that they sell at least 8 Type I bouquets each day, so they always make at least 8 of these each day. There are only 60 dozen tulips and 100 dozen irises available each day. How many of each type of bouquet should the shop make each day to maximize profit? 

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MATH2003 - Calculus and Linear Algebra I

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Introduces students to the application of computational and numerical methods for simulation, optimisation, linear and non-linear equations, differentiation and integration. Develops skills in applying numerical methods for solving application problems, utilising software such as Microsoft Excel and Visual Basic for Applications programming.

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Module 1:  Mathematical programming

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utilise Excel and Visual Basic for Applications to implement computational and numerical methods for solving problems

formulate and solve mathematical programming problems

implement numerical algorithms for computing solutions of problems such as non-linear and linear equations, numerical differentiation and numerical integration

demonstrate proficiency in the application of simulations, such as Monte Carlo simulations.

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Handbook of Formal Optimization pp 1–26 Cite as

Solving Cropping Pattern Optimization Problems Using Robust Positive Mathematical Programming

• Mostafa Mardani Najafabadi 3 &
• Somayeh Shirzadi Laskookalayeh 4  
• Living reference work entry
• First Online: 06 February 2024

22 Accesses

Agricultural activities occur in an environment that is constantly changing. In each cropping season, farmers must make management decisions based on numerous factors, some of which are beyond their control and others which are not. The use of mathematical programming models in determining optimal decisions for farmers, predicting the outcomes of policy effects, and the occurrence of uncontrollable factors in the agriculture sector is beneficial. It can provide planners and farmers with appropriate awareness and understanding of the effects of each decision related to resource allocation and cropping patterns before implementing that decision. One of the problems with some cropping pattern models is the consideration of resource amounts as fixed and certain, neglecting the issue of uncertainty. This results in a significant difference between the estimated model and the behavior of the farmers. In this context, the formulation of a mathematical programming model aligned with the real world and considering its uncertainties is highly important. This chapter aims to present an appropriate mathematical programming model for decision-making in determining cropping patterns and optimal resource allocation. This model should be able to model the uncertainties of the real world in the agriculture sector in the best possible way and provide more desirable and practical results. Therefore, while discussing the generalities related to the features and application of mathematical programming models and their types, the chapter elaborates on the basic and extended models of policy analysis using Positive Mathematical Programming (PMP) and the Robust Optimization (RO) approach. Subsequently, the method and a practical example of the combined model of Robust Positive Mathematical Programming (RPMP) in solving cropping pattern optimization problems is explained.

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Department of Agricultural Economics, Faculty of Agricultural Engineering and Rural Development, Agricultural Sciences and Natural Resources University of Khuzestan, Mollasani, Iran

Mostafa Mardani Najafabadi

Department of Agricultural Economics, Faculty of Agricultural Engineering, Sari Agricultural Sciences and Natural Resources University, Sari, Iran

Somayeh Shirzadi Laskookalayeh

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Correspondence to Mostafa Mardani Najafabadi .

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Institute of Artificial Intelligence, MIT World Peace University, Pune, Pune, Maharashtra, India

Anand J. Kulkarni

Faculty of Engineering and IT, University of Technology Sydney, Ultimo, NSW, Australia

Amir H. Gandomi

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Institute of Artificial Intelligence, Dr Vishwanath Karad MIT World Peace University, Pune, India

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Mardani Najafabadi, M., Shirzadi Laskookalayeh, S. (2024). Solving Cropping Pattern Optimization Problems Using Robust Positive Mathematical Programming. In: Kulkarni, A.J., Gandomi, A.H. (eds) Handbook of Formal Optimization. Springer, Singapore. https://doi.org/10.1007/978-981-19-8851-6_52-1

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DOI : https://doi.org/10.1007/978-981-19-8851-6_52-1

Received : 18 November 2023

Accepted : 25 November 2023

Published : 06 February 2024

Publisher Name : Springer, Singapore

Print ISBN : 978-981-19-8851-6

Online ISBN : 978-981-19-8851-6

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