Physical Models and Equilibrium Methods in Programming and Economics

I: Equilibrium of mechanical systems with linear constraints and linear programming problems.- 1.1. Introduction.- 1.2. Linear equations and inequalities.- 1.3. Systems of linear equations and inequalities.- 1.4. Linear programming problems. Duality theorems.- II: Equilibrium of physical systems and linear programming problems.- 2.1. Introduction.- 2.2. Some concepts from thermodynamics.- 2.3. Physical models of dual pairs of systems of linear equations and inequalities. Alternative theorems.- 2.4. A physical model for linear programming problems. Equilibrium conditions.- 2.5. Penalty methods.- 2.6. Some properties of approximate solutions of dual problems of linear programming problems.- 2.7. Models for transport type problems.- III: The method of redundant constraints and iterative algorithms.- 3.1. Introduction.- 3.2. The method of redundant constraints.- 3.3. The first iterative algorithm for solving linear programming problems and for solving systems of linear equations and inequalities.- 3.4. The second algorithm.- 3.5. Reduction of the general linear programming problem to a sequence of inconsistent systems. The third algorithm.- IV: The principle of removing constraints.- 4.1. Introduction.- 4.2. The method of generalized coordinates.- 4.3. The method of multipliers.- 4.4. Elastic constraints. Penalty function methods.- 4.5. Discussion.- V: The hodograph method.- 5.1. Introduction.- 5.2. The hodograph method for linear programming problems.- 5.3. Solution of the dual problem.- 5.4. Results of numerical experiments.- VI: The method of displacement of elastic constraints.- 6.1. Introduction.- 6.2. The first algorithm.- 6.3 The second algorithm.- 6.4. Combining the algorithms.- VII: Decomposition methods for linear programming problems.- 7.1. Introduction.- 7.2. Decomposition algorithms.- 7.3. Allocation of resources problems.- VIII: Nonlinear programming.- 8.1. Introduction.- 8.2. The principle of virtual displacements and the Kuhn-Tucker theorem.- 8.3. Numerical methods for solving nonlinear programming problems.- IX: The tangent method.- 9.1. Introduction.- 9.2. Constrained minimization problems.- 9.3. Linear programming.- 9.4. Dynamic problems of optimal control.- X: Models for economic equilibrium.- 10.1. Introduction.- 10.2. Equilibrium problems for linear exchange models.- 10.3. An algorithm for solving numerically equilibrium problems for linear exchange economies.- 10.4. Discussion. The Boltzmann principle.- 10.5. Equilibrium of linear economic models.- 10.6 Physical models for economic equilibrium. The equilibrium theorem.- 10.7. An algorithm for solving equilibrium problems for linear economic models.- 10.8. A generalization of the economic equilibrium problem.- XI: Dynamic economic models.- 11.1. Introduction.- 11.2. The Von Neumann-Gale model. Growth rates and interest rates.- 11.3. A method for solving the problem of maximum growth rates.- 11.4. Duality and problems of growth rates and interest rates.- 11.5. The minimal time problem.- 11.6. A time optimal control problem economic growth.- 11.7. A physical model for solving optimal control problems.- 11.8. Decomposition for time optimal control problems.- 11.9. Optimal balanced growth problems.- XII: Optimal control problems.