Physical Models and Equilibrium Methods in Programming and Economics

It isn't that they can't see Approach your problems from the solution. the right end and begin with It is that they can't see the the answers. Then one day, problem. perhaps you will find the final qu~stion. G. K. Chesterton. The Scandal of Father Brown ITh~ Point of 'The Hermit Clad in Crane Feathers' in R. van Gulik's a Pin'. The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. HowQvQr, thQ "tree" of knowledge of mathematics and related field does not grow only by putting forth new branches. It also happ~ns, quit~ often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathe­ matics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

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.