OPTIMIZATION METHODS
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1 D. Nagesh Kumar Associate Professor Department of Civil Engineering, Indian Institute of Science, Bangalore nagesh@civil.iisc.ernet.in URL: Brief Contents Overview of optimization and classification of optimization problems; Optimization using calculus, Kuhn-Tucker Conditions; Linear - Graphical method, Simplex method, Revised simplex method, Sensitivity analysis, Examples of transportation, assignment, water resources and other applications; Dynamic - Introduction, Sequential optimization, computational procedure, curse of dimensionality, Applications in water resources and structural engineering; Other topics in Optimization - Piecewise linear approximation, Multi objective optimization, Multi level optimization; Direct and indirect search methods; Evolutionary algorithms for optimization and search; Applications in civil engineering.
2 DETAILED CONTENTS : Introduction and Basic Concepts (05) Historical Development; Engineering applications of Optimization; Art of Modeling; Objective function; Constraints and Constraint surface; Formulation of design problems as mathematical programming problems; Classification of optimization problems based on nature of constraints, structure of the problem, deterministic nature of variables, separability of functions and number of objective functions; Optimization techniques classical and advanced techniques. : Optimization using Calculus (0) Stationary points - maxima, minima and saddle points; Functions of single and two variables; Global Optimum; Convexity and concavity of functions of one and two variables; Optimization of function of one variable and multiple variables; Gradient vectors; Examples; Optimization of function of multiple variables subject to equality constraints; Lagrangian function; Optimization of function of multiple variables subject to equality constraints; Hessian matrix formulation; Eigen values; Kuhn- Tucker Conditions; Examples. 3: Linear (0) Standard form of linear programming (LP) problem; Canonical form of LP problem; Assumptions in LP Models; Elementary operations; Graphical method for two variable optimization problem; Examples; Motivation of simplex method, Simplex algorithm and construction of simplex tableau; Simplex criterion; Minimization versus maximization problems; Revised simplex method; Duality in LP; Primal-dual relations; Dual Simplex method; Sensitivity or post optimality analysis; other algorithms for solving LP problems Karmarkar s projective scaling method. 4: Linear Applications (04) Use of software for solving linear optimization problems using graphical and simplex methods; Examples for transportation, assignment, water resources, structural and other optimization problems. 5: Dynamic (04) Sequential optimization; Representation of multistage decision process; Types of multistage decision problems; Concept of sub optimization and the principle of optimality; Recursive equations Forward and backward recursions; Computational procedure in dynamic programming (DP); Discrete versus continuous dynamic programming; Multiple state variables; curse of dimensionality in DP.
3 : Dynamic Applications (0) Problem formulation and applications for Design of continuous beam, Optimal geometric layout of a truss, Water allocation as a sequential process, Capacity expansion, Reservoir operation etc. 7: Integer (03) Integer linear programming; Concept of cutting plane method; Mixed integer programming; Solution algorithms; Examples. 8: Further topics in Optimization (0) Piecewise linear approximation of a nonlinear function; Multi objective optimization Weighted and constrained methods; Multi level optimization; Direct and indirect search methods; Evolutionary algorithms for optimization and search; Applications in civil engineering.
4 Lecture Plan. Introduction and Basic Concepts. Optimization using Calculus 3. Linear 4. Linear Applications Hours for Sub- Sub- Historical Development; Engineering applications of Optimization; Art of Modeling Objective function; Constraints and Constraint surface; Formulation of design problems as mathematical programming problems Classification of optimization problems Optimization techniques classical and advanced techniques. Stationary points; Functions of single and two variables; Global Optimum Convexity and concavity of functions of one and two variables Optimization of function of one variable and multiple variables; Gradient vectors; Examples. Optimization of function of multiple variables subject to equality constraints; Lagrangian function. Optimization of function of multiple variables subject to equality constraints; Hessian matrix formulation; Eigen values. Kuhn-Tucker Conditions; Examples. Standard form of linear programming (LP) problem; Canonical form of LP problem; Assumptions in LP Models; Elementary operations. Graphical method for two variable optimization problem; Examples. Motivation of simplex method, Simplex algorithm and construction of simplex tableau; Simplex criterion; Minimization versus maximization problems. Revised simplex method; Duality in LP; Primal-dual relations; Dual Simplex method; Sensitivity or post optimality analysis. Other algorithms for solving LP problems Karmarkar s projective scaling method. Use of software for solving linear optimization problems using graphical and simplex methods. Examples for transportation, assignment, water resources, structural and other 3 optimization problems. Total Hours 5 4
5 5. Dynamic. Dynamic Applications 7. Integer 8. Advanced Topics in Optimization Sub- Sequential optimization; Representation of multistage decision process; Types of multistage decision problems; Concept of sub optimization and the principle of optimality. Recursive equations Forward and backward recursions; Computational procedure in dynamic programming (DP). Discrete versus continuous dynamic programming; Multiple state variables; curse of dimensionality in DP. Problem formulation and application in Design of continuous beam and Optimal geometric layout of a truss. Hours for Sub- Water allocation as a sequential process Capacity expansion and Reservoir operation Integer linear programming; Concept of cutting plane method. Mixed integer programming; Solution algorithms; Examples. Piecewise linear approximation of a nonlinear function Multi objective optimization Weighted and constrained methods; Multi level optimization. Direct and indirect search methods. Evolutionary algorithms for optimization and search. Applications in civil engineering. Total Hours 4 3 Total 40
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