(558c) Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs | AIChE

(558c) Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs


Zavala, V. M. - Presenter, University of Wisconsin-Madison
Shin, S., Uninveristy of Wisconsin-Madison
Anitescu, M., Argonne National Laboratory
A graph-structured nonlinear program (gNLP) is a nonlinear optimization problem whose algebraic structure is induced by a graph. These problems arise in diverse applications such as dynamic optimization (model predictive control and moving horizon estimation) [1,2], network optimization (energy systems and supply chain) [3,4,5], optimization with embedded discretized partial differential equations (PDEs) [6], and multi-stage stochastic programming [7]. Characterization of the nodal sensitivity of such problems (the sensitivity of the solution at a particular node against the data perturbation on another node) allows addressing a wide range of questions that are of practical interest. For example, in the context of dynamic optimization, it allows understanding the relationship between the prediction horizon length and the control/estimation performance [8]. This further enables systematically balancing the computational complexity and the expected control/estimation performance. Furthermore, in the context of energy infrastructures, it allows understanding how far the disturbance in one part of the network may propagate to the other part; this enables assessing the flexibility and reliability of operations [9].

In this work, we aim to quantitatively characterize the nodal solution sensitivity of gNLPs. Building upon the existing NLP sensitivity theory [10,11], we show that the nodal sensitivity decays exponentially with respect to the distance to the perturbation point [12]. Remarkably, this result (which we call exponential decay of sensitivity; EDS) holds under fairly standard assumptions used in classical NLP sensitivity theory: second-order sufficiency conditions (SOSC) and the linear independence constraint qualification (LICQ) [10,11]. In practical terms, these regularity conditions (LICQ and SOSC) can be interpreted as having sufficient flexibility and positive objective curvature. This interpratation enables an intuitive explanation of EDS that the positive objective curvature and the flexibility of the system damps the propagation of the perturbation. In the context of control, the regularity conditions can be obtained from controllability and observability [13]. These results generalize our previous results under simpler settings [14,15]. Also, it provides new insights on how perturbations propagate through graphs and on how the NLP formulation influences such propagation. Moreover, EDS allows the creation of novel computing strategies and optimization formulations, which include the overlapping Schwarz decomposition method (also known as domain decomposition) [14,16,17], online algorithms for dynamic optimization [18], diffusing-horizon model predictive control [19], and flexibility-maximizing design method (currently under study).


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