Linear Programming and Associated Functions to Building a Linear Model (Statistics Project Sample)

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Introduction
Optimization is a concept in linear programming (LP) which can be defined as an act or rendering optimal equations required for the profitability and growth in a linear programming problem. In this concept, researchers seek to find variables that lead to optimal values in a function to be optimized (Karloff, 1991). In the problems relating to optimization, unique software called an optimization toolbar can be used. This requires that only the experts can make use of it and not the non-skilled. This software provides a wide range of algorithms in standard large-scale optimization.
To add on this, the software contains a program use to solve the constrained and the non-constrained variables, the discrete and the continuous variables by use of installed functions. The functions contained in this software include those of finding binary integers in LP, quadratic programming functions, the linear and non-linear functions, and the multi-objective functions, which can be used to find optimal solutions and even perform a tradeoff analysis. They in addition, can be applied in finding balance multiple designs; incorporate methods in the algorithms section and develop superb models to help in solving the optimization problems (Powell & Baker, 2004).). In this essay discussion, the main issue that will be discussed is the use of linear and non-linear optimization techniques. Contrast between the non-linear and linear optimization shall be featured and finally a scenario that will be used to obtain the best answer shall also be considered.
Discussion
Linear Optimization
With linear optimization problems, the researcher is interested in determining the function or a variable to be either minimized or maximized. This is done in the subject to the given defined constraints called the set of alternatives. Functions that are supposed to be maximized or minimized are called the objective functionswhile the set of alternatives iscalled the feasible regions or in other words, the constraint region. Such regions are taken to take the subset Rn–real dimensional space. The objective function region ranges from Rnto R. Sometimes, it is recommended to restrict a class of the optimization problem over Rn especially where the objective function is linear and has the form of ax1 + ax2+ …+cnxn. Linear programming is one of the best tools that can be used to address issues on optimization. It has many areas of application such as in production scheduling, warehousing, layout, allocation of facilities, flight crew, portfolio optimization, parameter estimation, and allocation of resources.
An example of a way to set up a linear programming problem is simple and needs to be followed to the dot by anybody interested in solving the LP. The first step is to identify the decision variables and label them appropriately. The identification of the decision variables will help the researcher determine what is required in case of any adjustments are required. The second is that of identifying the objective and apply the decision variables in writing an expression for the main function. The next step is to determine the explicit constraints and write an expression in a functional manner to express each of the constraints. Lastly, it will be recommended to determine functions in their implicit forms as the solution is to be found.
The general function in the linear programming problem can be presented as follows. For instance, in this case, a maximization problem is to be considered. The counter to the maximization problem, a minimization problem can also be inferred since nothing changes much from the maximization problem: Max ax1 + ax2+ ax3+… +anxn≤bxn
St.
Sx1+ sx2-sx3≤ txn
Wx1+ wx2+wx3≤pxn
X≥0 (the non-negativity constraint)
The corresponding minimization problem can be rewritten as an example as shown below,
Minimize cx1+ cx2+… cnxn
Subject to a11x1+ ax12x2+ …+a1nxn≤ b1
a21x1+ a22x2+…a2nxn≤b2
….
av1x1+av2x2+… avnxn≤bv
In this optimization, the x1….xnare called the decision variables while the v-variables are the inequality constraints.
Characteristics of Linear Optimization Functions
The following is some of the characteristics of the linear programming functions. One, the objective function, may be of minimization type for instance, f(x1, x2,x3..Xn) is equivalent to the minimization of same functions and can be rewritten as minimization form in an optimization problem. The second point is that, all the functions are of equality types. When this happens, the functions can be expressed to form equations from the inequality signs by adding the slack variables if it is a less than function. For the greater than or equal to functions, they can be converted to equality functions by subtracting the slack variables from the inequality functions. The third point is that, most of the engineering problems will have decision variables representing the physical dimensions and with that, the xj values may be non-negative. However, in such cases, the variables can be restricted in their signs and be written as the difference between the two variables.
Non-Linear Optimization
In non-linear optimization problem, the variables are non-linear, and the parameter values are constrained to some regions.By use of mathematical formulae, the issues can be solved with ease and obtain a variety of optimization techniques. With non-linear problems, the prime objective is to find the largest value of a certain function for instance, y2-m and determine whether it is within the region of unit. For example, max [{y2- r, x2- t2≤ 1},{x,t}].
In non-linear programming optimization toolbar, it provides algorithms for solving these problems in the MATLAB. The bar contains solvers that are unconstrained and others constrained nonlinear optimizations and helps to solve for the least square optimizations. The bar uses the three algorithms to solve the unconstrained minimization of non-linear problems; the Nelder-Mead called the downhill simplex. It uses only the direct algorithm searches and handles the non-smooth functions, the Quasi-Newton algorithms is mixed with quadratic and the cubic line functions and the search procedures in updating the calculation in the Hessian Matrix, and lastly the tool bar uses the trust region for the unconstrained problems of non-linear optimization. This concept is useful in hefty scale problems in which, sparsity or the structures can best be exploited.
The use of non-linear functions may be accompanied by a lot of functions and applications that may require a lot of expertise in...