## Aggregation in Large-Scale OptimizationWhen analyzing systems with a large number of parameters, the dimen sion of the original system may present insurmountable difficulties for the analysis. It may then be convenient to reformulate the original system in terms of substantially fewer aggregated variables, or macrovariables. In other words, an original system with an n-dimensional vector of states is reformulated as a system with a vector of dimension much less than n. The aggregated variables are either readily defined and processed, or the aggregated system may be considered as an approximate model for the orig inal system. In the latter case, the operation of the original system can be exhaustively analyzed within the framework of the aggregated model, and one faces the problems of defining the rules for introducing macrovariables, specifying loss of information and accuracy, recovering original variables from aggregates, etc. We consider also in detail the so-called iterative aggregation approach. It constructs an iterative process, at· every step of which a macroproblem is solved that is simpler than the original problem because of its lower dimension. Aggregation weights are then updated, and the procedure passes to the next step. Macrovariables are commonly used in coordinating problems of hierarchical optimization. |

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### Contents

Aggregated Problem and Bounds for Aggregation | 1 |

11 Definitions and Preliminary Results | 2 |

12 A Posteriori Bounds | 6 |

13 A Priori Bounds | 8 |

14 Some Generalizations | 10 |

142 A Priori Bounds | 17 |

15 Illustrative Example and Numerical Indications | 19 |

2 The Generalized Transportation Problem | 25 |

8 Distributed Parameter Systems | 135 |

9 Numerical Indications | 147 |

References to Chapter 2 | 157 |

Consistent Aggregation in Parametric Optimization | 163 |

2 Parametric Linear Programming | 166 |

21 The Base Problem | 167 |

22 Aggregation of Variables | 170 |

23 Constraint Reduction | 173 |

21 The Aggregated Problem | 26 |

22 The Error Bounds | 30 |

23 Numerical Indications | 35 |

3 Variable Aggregation in Nonlinear Programming | 38 |

4 Sharpening Localization by Valid Inequalities in Integer Programming | 44 |

5 The Generalized Assignment Problem | 51 |

References to Chapter 1 | 58 |

Iterative AggregationDecomposition in Optimization Problems | 61 |

1 Linear Aggregation in FiniteDimensional Problems | 62 |

2 Problems with Constraints of Special Structure | 72 |

3 Decomposition of the Macroproblem in the BlockSeparable Case | 92 |

4 Adaptive Clustering in Variable Aggregation | 104 |

5 Aggregation of Constraints | 112 |

6 Aggregation of Controls in Dynamic Problems with Mixed Constraints | 121 |

7 Aggregation of State Variables in LinearQuadratic Problems | 129 |

24 The Aggregated Problem in the Standard Form and the Proof of Its Consistency | 177 |

25 The Degenerate Case | 187 |

26 Various Forms of the Base Problem | 189 |

27 An Example of Varying One Element in the Constraint Matrix | 198 |

3 Parametric Convex Programming | 202 |

32 Parameters in Convex Constraints | 217 |

4 Special Classes of Problems | 233 |

42 LinearConvex Problems in Banach Spaces | 245 |

43 Optimal Control of Linear Systems under Terminal Constraints | 256 |

44 Weaker Consistency Conditions Integer Linear Programming | 264 |

441 A QuasiConsistency Theorem | 272 |

442 Using the Cuts of the Primal Algorithm | 278 |

Comments and References to Chapter 3 | 284 |

289 | |

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### Common terms and phrases

affine manifold aggregated variables aggregation method aggregation parameters aggregation technique algorithm apply approach ascent direction assume assumption Banach space base problem basic variables calculated cluster coefficients columns components Comput consider consistent aggregation constraints construct convex programming corresponding decomposition defined definition denote derivatives differentiable dimension disaggregated solution dual problem dual solution equations error bounds feasible solution independent subproblems integer integer linear programming integer programming iterative aggregation knapsack problem Lagrange multiplier Lemma linear programming problem linearly independent Litvinchev I.S. localization macroproblem matrix nonbasic nondegenerate nonnegative nonunique objective function obtain optimal control problems optimal objective value optimal value optimization problems parameter variations parametric problem partially aggregated problem primal priori bound resource saddle point satisfies scalar Section Sm+i solved Step stepsize Suppose UB(u upper bound valid inequality vector weights Zipkin