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**Linear Programming**

Linear programming refers to a mathematical approach which observes the given criteria of optimality for allocating the available resources to the appropriate activities. LP is a technique which is used to mitigate the cost function and maximize the profit function, revenue and contribution margin.

LP enables user to solve the numerous complex problems without using various types of techniques. It used in the various areas like business, military, industries, economics and many more. Linear Programming deals with the various concepts viz. Integer linear programming, Network flow optimization, Linear-fractional optimization, Integer programming, Primal-dual interior-point method, Maximum and Minimum analysis, etc.

The main problem that occurs in the Linear Programming is that it has numerous constraints. It is the mostly used form of constrained optimization which is difficult than the unconstrained optimization. Main components that involved in the Constrained Optimization problem are as follows:

- Decision Variable: it shows the determined quantities.
- Constraints: it represents the restricted quantities resources which are used by the decision variables. For expressing the limits on the possible solutions, it uses variables simultaneously.
- Objective function: it shows the consequence of the decision variables on the cost to be optimized. It finds the result by combining variables in it.
- Data quantifies relationship: it mainly determines the constraints and the objective functions.

On the basis of variables, Duality theory is used in linear programming to establish the relationship between two linear programs. These variables are used with the shadow-price interpretation which further elaborated in two ways.

- First, to know about the linear-programming model, you have to know about the shadow-price interpretation of the optimal simplex multipliers and then by using computational efficiencies, one can solve the problem of linear programming with the shadow prices in the form of variables.
- Duality theory is a unifying theory, problems of the Linear Programming is translated into the dual problem. A linear program can be of two types, either feasible or unbounded. By using duality theory, one can perfectly determine the linear program. If the dual is infeasible then primal will be unbounded and if the dual is unbounded then primal will be infeasible.

Moreover, A Method which is used for solving the all Linear Programming problems is termed as Simplex method. It is used for solving the problems comes with a complex steps that is named as Pivot selection. Pivot selection enables one to choose the variables to be entered in a linear program. Good selected variables lead to the fast progress in program while bad selected variables are responsible for making the progress slow. Three majorly used pivot heuristics rules are as follows:

- Dantzig rule: this heuristic rule choose the negative multiplier variables to be entering into the column.
- Devex: for find the optimal solution, this rule recognizes the Steepest Edge in its search.
- Steepest Edge: in an objective, Steepest Edge selects the variable according to the largest rate of decrease.

Complexity of LP involves the simplex methods and LP-completeness. Few problems of the LP need O (n log n) operations with computation of the convex hulls subsets. The resultant output produces the NP-compete version of LP problems. To find the optimal solution for LP problems, simplex method needs exponential number of iterations. This can be done by creating a polotype with exponential number of iterations.

Ellipsoid algorithm is an application of ellipsoid methods which is used for the Linear Programming. Polynomial time algorithm in Linear Programming can be derived by using ellipsoid method. Ellipsoid algorithm is not much better than the Simplex algorithm. It is better than Simplex methods in theoretical way but not good in the practical way. It is majorly used for developing the polynomial time algorithm and also used for solving the feasibility problems. If a problem has SepOracle procedure then it can be solved by using Ellipsoid algorithm. It can also used for solve the large convex optimization problems.

Furthermore, Linear programming is a wide area of study, with the upper explained concepts, it involves many more concepts, including Structural optimization, The central path, Barrier method, equilibrium theorem, Inventory control problems, Self-dual formulations, etc. There are various advanced concepts that fall under the Linear Programming field. These advanced concepts mostly choose by the PhD level students to completing their thesis and research report. These advanced concepts are listed below:

- Dual Simplex Method
- Matrix generators and modelling languages
- Dual linear program
- Parametric analysis
- Scaling
- Decomposition in Linear Programming
- Perturbation function and sensitivity

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**Some of the homework help topics include :**

- Networks and the shortest path problem,integer linear programs,LP duality and game theory,Maximin and minimax objectives,An economic interpretation of LP duality ,Weak and strong duality,Bounds and the dual LP,Degeneracy, convergence, multiple optimal solutions,The simplex method ,Linear programs in canonical form,Geometry and algebra of "corner points",Improving search: convexity and optimality,algorithm design,Resource allocation models,Sets, summations, for statements,Production process models
- Blending models,Resource allocation models,optimization models with a computer,Graphical solutions of optimization models,optimization modeling,production and marketing, simplex techniques, duality theory, parametric analysis, Wolfe and Dantzig's ,decomposition methods, ellipsoid method,geometry of 2-dimensional linear programs,Simplex methodfeasible solutions,pivoting,degeneracy,cycling and anti-cycling pivot rule,phase I and phase II algorithms,duality ,sensitivity analysis,introduction to computational complexity
- ellipsoid method,interior point methods,network flow problems,,The Standard Maximum and Minimum Problems,The Diet Problem ,The Transportation Problem ,The Activity Analysis Problem ,The Optimal Assignment Problem,Terminology,Dual Linear Programming Problems ,The Duality Theorem ,The Equilibrium Theorem,Interpretation of the Dual,The Pivot Operation,The Simplex Method,The Simplex Tableau ,The Pivot Madly Method ,Pivot Rules for the Simplex Method ,The Dual Simplex Method,Generalized Duality ,The General Maximum and Minimum Problems
- classification of optimization problems

formulation of linear and integer programs

available software to solve linear programs

geometry of 2-dimensional linear programs

Simplex method

feasible solutions

pivoting

degeneracy

cycling and anti-cycling pivot rule

phase I and phase II algorithms

duality

sensitivity analysis

introduction to computational complexity

ellipsoid method

interior point methods

network flow problems

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- Support vector machine (SVM) problems,convex hull decision problem and SVM,Linear inequalities and the feasibility problem,linear programming problem (LP),Strict feasibility problem,Dantzig’s simplex method,Feasible Solutions,Degeneracy, cycling,Bland’s rule for finite termination,Worst-case complexity of LP,Klee-Minty ,Hirsh conjecture,Sensitivity analysis and parametric LP,Farkas lemma,Gordan theorem,geometric interpretations,Duality in linear programming,Lagrange multipliers and Karush-Kuhn-Tucker optimality conditions
- Duality theorems,Complementary slackness conditions,fundamental theorem of LP,Dual of LP in Standard form ,dual simplex method,The primal-dual method for LP ,Convex sets,Polyhedra and polyhedral cones,Caratheodory, Farkas-Minkowski-Weyl,Helly theorems,Game theory and von Neumann’s min-max theorem,The triangle algorithm,Khachiyan’s ellipsoid method for LP,Karmarkar’s algorithm and variations,Taylor theorem,Strongly polynomial-time algorithms,Total unimodularity and structured LP,shortest paths,mean cycles,max flows,bipartite matching,min-cost flows,multicommodity flows,spanning tree,general weighted matching problem,TSP and magic labeling problem
- Standard Maximum and Minimum Problems

The Diet Problem

The Transportation Problem

The Activity Analysis Problem

The Optimal Assignment Problem

Terminology

Dual Linear Programming Problems

The Duality Theorem

The Equilibrium Theorem

Interpretation of the Dual

The Pivot Operation

The Simplex Method

The Simplex Tableau

The Pivot Madly Method

Pivot Rules for the Simplex Method

The Dual Simplex Method

Generalized Duality

The General Maximum and Minimum Problems

Solving General Problems by the Simplex Method

Solving Matrix Games by the Simplex Method

Cycling

A Modification of the Simplex Method That Avoids Cycling

Four Problems with Nonlinear Objective Function

Constrained Games

The General Production Planning Problem

Minimizing the Sum of Absolute Values

Minimizing the Maximum of Absolute Values

Chebyshev Approximation

Linear Fractional Programming

Activity Analysis to Maximize the Rate of Return

The Transportation Problem

Finding a Basic Feasible Shipping Schedule

Checking for Optimality

The Improvement Routine - Decision variables

objective function

bounds and constraints

The feasible region

geometric and algebraic characterisation

dual of an LP problem and duality theory

Theory underlying sensitivity and fair prices

Modelling

Xpress mathematical programming system

means of modelling, solving and analysing LP case studies

Exploration of the modelling language Mosel

index sets, data arrays, decision variables, constraints

simplex algorithm for LP problems

Geometric and algebraic concepts

Proof of termination for non-degenerate LPs

Linear algebra underlying its implementation

revised simplex method - Graphical solutions and sensitivity in two-dimensions

Interpreting the output of optimization software

simplex algorithm

Sensitivity analysis

Duality

Minimum cost network flow

modeling and applications

min cost network flow

Shortest path and dynamic programming

Project management