Algorithms for diode-source networks 56—71
Algorithms for linear programming 101—107
Algorithms for network flow problems 73—74
Algorithms for quadratic programming 104—107
Algorithms for the maximum flow problem 72—73
Algorithms for the shortest path problem 71—72
Basic solution of generalized terminal-pair system 171
Basic solution of terminal-pair system 81
Breakpoint curves as solution set of a terminal-pair system 159—168
Breakpoint curves as terminal characteristic of networks containing diodes 39—43
Breakpoint curves, defined 159—160
Breakpoint tracing method for a reduced terminal-pair system 91—92
Breakpoint tracing method for a terminal-pair system 82—87
Breakpoint tracing method for an uncoupled system 88—91
Breakpoint tracing method for the completely degenerate system 92
Breakpoint tracing method for the generalized system 87—88 169—180
By-pass method for linear and quadratic programming 103—106
By-pass method for obtaining feasibility 98—101
Complemementary slackness and the diode circuit element 34
Complemementary slackness in general programming 12—17
Complemementary slackness principle, for concave programming 157
Complemementary slackness principle, for linear and quadratic programming 25 143—144
Concave (convex) functions 9—11 148—152
Concave (convex) functions, defined 148
Concave (convex) functions, properties 148—152
Concave (convex) programming 26 38—39 153—158
Concave (convex) programming and nonlinear resistors 38—39
Concave (convex) programming, complementary slackness 157
Concave (convex) programming, duality 26 157
Concave (convex) programming, Lagrangian problem 154
Cone, defined 12
Conjugate variable pairs 81
Constraint qualifications 132—133 135—137
Constraint set 7
Convex hull, defined 120
Convex set, defined 120
Convex set, defined by convex inequalities 10 131
Convex set, polyhedral 121
Current reduced network 34—35 56 93 96—101
Current source 29
Degeneracy in the breakpoint tracing method 81 87 169
Diode and complementary slackness 34
Diode, defined 29—31
Diode-source networks, algorithms for solving 56—71
Diode-source networks, existence conditions 54—55
Diode-source networks, nonredundancy assumptions 53—54
Duality and conservation of energy 34 93
Duality in ordinary minimization 21—23
Duality theorem, for concave programming 157
Duality theorem, for linear and quadratic programming 23—25 141
Duality, topological, for diode-source networks 51
Electrical devices, current sources 29
Electrical devices, diodes 29—31
Electrical devices, resistors, linear 31
Electrical devices, resistors, nonlinear 38—39
Electrical devices, transformers 31
Electrical devices, voltage sources 29
Electrical models for linear programming 93—95
Electrical models for maximum flow problem 47—48
Electrical models for network flow problem 51—52
Electrical models for quadratic programming 93—95
Electrical models for shortest path problem 49—51
Electrical models for terminal-pair system 78—80
Electrical networks, diode-source 53—71
Electrical networks, diode-source-resistor 31—34
Electrical networks, graphs 27
Electrical networks, incidence matrix 27—28
Electrical networks, laws 29
Electrical networks, reduced 34—35 56—62 67—68 93 96—101
Electrical networks, with ideal transformers 35—37
Existence conditions for diode-source networks 54—56
Existence theorem for linear and quadratic programming 142
Existence theorem, interpretation in terms of reduced networks 34—35
Extreme point, defined 120
Feasible vector 6
Ford — Fulkerson method 73—74
Functions, concave (convex) 9—11 148
Functions, Legendre transformation 18—21 148—152
Functions, strict increase property 148
General programming problem 6—7 9—17 113—117 131—137
General programming problem, complementary slackness 12—17
General programming problem, constraint qualification 132—133 135—137
General programming problem, fundamental theorem 17 133—134
General programming problem, Kuhn — Tucker conditions 12—17 133—134
General programming problem, Lagrange multipliers 11—17
General programming problem, Lagrangian problem 11—17 133
General programming problem, local and global minima 9—11 131—132
General programming problem, method of steepest descent for 113—117
Generalized terminal-pair system 87—88 169—180
Gradient method for equality constrained minimization 110—113
Gradient method for the general programming problem 113—117
Gradient method for unconstrained minimization 109—110
| Graph of a network 27—28
Half space, defined 119
Hyperplane, defined 119
Incidence matrix 27—28
Incremental solutions of terminal pair system 82—83
Kirchoff’s loop law 29
Kirchoff’s node law 29
Kuhn — Tucker conditions 12—17 133—134
Lagrange multipliers 11—17
Lagrangian problem for concave programming 154
Lagrangian problem for equality constrained minimization 22
Lagrangian problem for linear and quadratic programming 24 139—149
Lagrangian problem for the general programming problem 11—17
Legendre transformation 18—21 148—152
Legendre transformation, defined 20 152
Legendre transformation, example 21
Legendre transformation, properties 148—152
Line, defined 12
Linear programming 7—8 23—25 93—95 101—107 138—147
Linear programming, by-pass algorithm 104—107
Linear programming, complementary slackness 25 143—144
Linear programming, duality 23—25 141
Linear programming, electrical model 93—95
Linear programming, existence of optima 25 141
Linear programming, valve algorithm 101—103
Local minima 9—10 109 131
Loop, in a network 53
Maximum flow problem, algorithm for 72—73
Maximum flow problem, electrical model 47—48
Minimization problems, duality in 21—23
Minimization problems, equality constrained, steepest descent method 110—113
Minimization problems, quadratic 21—23
Minimization problems, steepest descent method for 108—110
Monotone functions, strict increase property 148
Network flow problems 45—53 71—75
Network flow problems with quadratic costs 74—75
Network flow problems, electrical model 51—52
Network flow problems, Ford — Fulkerson algorithm 73—74
Notation 4—5
Objective function 6
Optimal vector 7
Path, in a network 53
Power 29
Power and duality 34 93
Primal-dual method 106
Programming, concave (convex) 26 38—39 153—158
Programming, general (nonlinear) 6—7 9—17 113—117 131—137
Programming, linear 7—8 23—25 93—95 101—107 138—147
Programming, quadratic 8 23—26 93—95 104—107 138—147
Quadratic programming 8 23—26 93—95 104—107 138—147
Quadratic programming, application to a steepest descent method for the general programming problem 114—117
Quadratic programming, by-pass algorithm 104—107
Quadratic programming, complementary slackness 25 143—144
Quadratic programming, duality 23—25 141
Quadratic programming, electrical models 93—95
Quadratic programming, existence of solutions 25 142
Quadratic programming, Lagrangian problem 24
Quadratic programming, uniqueness 24 143—144
Reduced networks and existence of solutions 35 54—56
Reduced networks and feasibility 34—35 93
Reduced networks, current 34—35 56 67—68 93 96—101
Reduced networks, voltage 34—35 56—62 93 101
Relative minima 9—11 109 131
Shortest path problem, algorithm for 71—72
Shortest path problem, electrical model 49—51
Simplex method 77—78 103
Steepest descent method for equality constrained minimization 110—113
Steepest descent method for nonlinear programming 113—117
Steepest descent method for unconstrained minimization 108—110
Strict increase property 148
Summary of results 2—3
Terminal pair system for linear programming 101—107
Terminal pair system for obtaining feasibility 96—101
Terminal pair system for quadratic programming 104—107
Terminal pair system, basic solutions 81
Terminal pair system, completely degenerate 92
Terminal pair system, degeneracy 81 87 169
Terminal pair system, dual reduced 92
Terminal pair system, electrical model for 78—79
Terminal pair system, generalized 87—88 169—180
Terminal pair system, incremental solutions 82—83
Terminal pair system, primal reduced 91—92
Terminal pair system, terminal solutions and break-point curves 39—43
Terminal pair system, tracing procedure 82—87
Terminal pair system, uncoupled 88
Terminal pair system, unit solutions 82—83
Transportation problem 46
Uniqueness of solutions in concave programming 157
Uniqueness of solutions in quadratic programming 25 143—144
Unit solutions of terminal-pair system 82—83
Valve algorithm for linear programming 101—103
Valve algorithm for obtaining feasibility 96—98 101
Voltage reduced network 34—35 56—62 93 101
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