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Linear Programming is an integral part of mathematical model which is used to find maximum profit or minimum cost through linear function in business. It is an optimization technique which is used to guide quantitative decisions for a linear objective function and system of linear constraints to achieve the best outcome whose requirements are represented by linear relationships in a mathematical model.
Few Topics are:
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- parametric programming , distribution problems , formulation and basic definitions , integer programming, cutting planes and branch , bound methods, software for linear programming.
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- Properties of Matrix Operations, Symmetric Matrices and Seriation in Archaeology, The Inverse of a Matrix and Cryptography, The Leontief Input-Output Model in Economics, Markov Chain
- Population Movements, and Genetics, Communication Model and Group Relations in Sociology.,,Determinants ,,Computing Determinants of 2×2 and 3×3 Matrices, Properties of Determinants
- Matrix Inverses and Systems of Linear Equations.,,General Vector Spaces ,,Vectors, Dot Product, Norm, Angle, and Distance,Vector Spaces and Subspaces, Linear Combination of Vectors
- Linear Dependence and Independence, Bases and Dimension of Vector spaces, Rank of a Matrix, Inner Product Space application, Least Squares Curves.,,Coordinate Representations Least Square Curves
- Numerical Methods, Gaussian Elimination, The Method of LU Decomposition.,,,,Linear Programming Concepts , The Simplex Method , Geometrical Explanation of the Simplex Method
- Linear functions and their graphs, Matrices, Linear systems of equations, Linear programming, Set theory, Counting techniques, Probability theory, Statistics, Finance problems
- Develop linear models, Theory , computational aspects , simplex method, Duality theory , sensitivity analysis, interior point methods , column generation, Multiobjective linear programs.
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Few topics for homework help:
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- Fundamental Theorem , Linear Programming, Weak and Strong Duality Theorems
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Few Topics Covered by Linear Programming
- Deterministic Models, Linear Programming, formulating optimization problems, spreadsheet modeling, Solver, SolverTable, sensitivity analysis, multi period modeling, Integer Programming
- Integer variables, binary variables, logical relationships, Network Optimization, transportation problem, minimum cost flow, funds flow model, project management, currency exchange, Non linear Optimization
- Portfolio optimization, demand estimation, pricing, Probabilistic Models, Stochastic Optimization, Newsvendor model, Simulation, Basic concepts, Crystal Ball software, option pricing
- Optimization , revenue management, liquidity optimization, Optimization, Linear Algebra, Vectors , Matrices, Vector Spaces, Linear Systems, Formulations of Linear Programs, Properties of Linear Functions, Properties of Linear Equality Constraints
- Optimality Conditions for Equality Constraints, Properties of Linear Inequality Constraints, Feasible directions , Vertices, Finding Vertices, Optimality Conditions for LP, Farkas Lemma
- Simplex Method, Properties of Simplex Method, Finding a feasible point, Linear Programs with mixed constraints, Linear Programs with Standard form, Solving LPs, Standard form, Basic Duality Theory
- Duality, Duality Inequality Form, Formulating Duals, Primal, Dual
- Linear programming problems,Systems of linear inequalities:,Farkas' lemma,Duality theorem of linear programming,Convex cones and polyhedra,The Decomposition theorem for polyhedra
- structure of polyhedra:,faces, facets and vertices,The geometric description of the simplex method,The simplex method in tableau form,The Bland's rule,The two-phase method,The geometric description of the dual simplex method
- The dual simplex method in tableau form,The transportation problem,The convex hull decision problem and its generalization: ,linear programming,support vector machine (SVM),distance duality and the triangle algorithm
- Linear inequalities and the feasibility problem,linear programming problem (LP),standard form,convex hull decision problem,strict feasibility problem,Dantzig’s simplex method
- Basic Feasible Solutions,Degeneracy, cycling,Bland’s rule for finite termination,Worst-case complexity of LP:,Klee-Minty examples,Hirsh conjecture,Sensitivity analysis and parametric LP;,Farkas lemma,Gordan theorem,geometric interpretations
- Algorithmic applications,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,The dual simplex method,The primal-dual method for LP,Convex sets,Polyhedra and polyhedral cones,extreme points,extreme directions
- recession directions,edges, facets, basic feasible solutions,Fourier-Motzkin elimination method,Representation theorems: ,Caratheodory,Farkas-Minkowski-Weyl,Helly theorems
- Game theory ,von Neumann’s min-max theorem,The triangle algorithm: ,fully polynomial-time approximation scheme,algorithmic separation of convex sets,The particular case of SVM;,Khachiyan’s ellipsoid method for LP
- Connections to the convex hull membership problem,Notions of size of LP,rounding, precision and polynomial-time algorithms,Karmarkar’s algorithm and variations,connections to the convex hull membership problem,Taylor theorem
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