The explicit linear quadratic regulator for constrained systems

We present CasADi, an open-source software framework for numerical optimization. CasADi is a general-purpose tool that can be used to model and solve optimization problems with a large degree of flexibility, larger than what is associated with popular algebraic modeling languages such as AMPL, GAMS, JuMP or Pyomo. Of special interest are problems constrained by differential equations, i.e. optimal control problems. CasADi is written in self-contained C++, but is most conveniently used via full-featured interfaces to Python, MATLAB or Octave. Since its inception in late 2009, it has been used successfully for academic teaching as well as in applications from multiple fields, including process control, robotics and aerospace. This article gives an up-to-date and accessible introduction to the CasADi framework, which has undergone numerous design improvements over the last 7 years.

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CasADi: a software framework for nonlinear optimization and optimal control

Since its popularization in the late 1970s, Sequential Quadratic Programming (SQP) has arguably become the most successful method for solving nonlinearly constrained optimization problems. As with most optimization methods, SQP is not a single algorithm, but rather a conceptual method from which numerous specific algorithms have evolved. Backed by a solid theoretical and computational foundation, both commercial and public-domain SQP algorithms have been developed and used to solve a remarkably large set of important practical problems. Recently large-scale versions have been devised and tested with promising results.

Sequential Quadratic Programming

Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

Model Predictive Control (MPC), the dominant advanced control approach in industry over the past twenty-five years, is presented comprehensively in this unique book. With a simple, unified approach, and with attention to real-time implementation, it covers predictive control theory including the stability, feasibility, and robustness of MPC controllers. The theory of explicit MPC, where the nonlinear optimal feedback controller can be calculated efficiently, is presented in the context of linear systems with linear constraints, switched linear systems, and, more generally, linear hybrid systems. Drawing upon years of practical experience and using numerous examples and illustrative applications, the authors discuss the techniques required to design predictive control laws, including algorithms for polyhedral manipulations, mathematical and multiparametric programming and how to validate the theoretical properties and to implement predictive control policies. The most important algorithms feature in an accompanying free online MATLAB toolbox, which allows easy access to sample solutions. Predictive Control for Linear and Hybrid Systems is an ideal reference for graduate, postgraduate and advanced control practitioners interested in theory and/or implementation aspects of predictive control.

Predictive Control for Linear and Hybrid Systems

In this paper we establish a theoretical basis for utilizing a penalty-function method to estimate sensitivity information (i.e., the partial derivatives) of a localsolution and its associated Lagrange multipliers of a large class of nonlinear programming problems with respect to a general parametric variation in the problem functions. The local solution is assumed to satisfy the second order sufficient conditions for a strict minimum. Although theoretically valid for higher order derivatives, the analysis concentrates on the estimation of the first order (first partial derivative) sensitivity information, which can be explicitly expressed in terms of the problem functions. For greater clarity, the results are given in terms of the mixed logarithmic-barrier quadratic-loss function. However, the approach is clearly applicable toany algorithm that generates a once differentiable "solution trajectory".

Sensitivity analysis for nonlinear programming using penalty methods

We give a brief overview of important results in several areas of sensitivity and stability analysis for nonlinear programming, focusing initially on “qualitative” characterizations (e.g., continuity, differentiability and convexity) of the optimal value function. Subsequent results concern “quantitative” measures, in particular optimal value and solution point parameter derivative calculations, algorithmic approximations, and bounds. Our treatment is far from exhaustive and concentrates on results that hold for smooth well-structured problems.

Sensitivity and stability analysis for nonlinear programming

Mathematical programming study 21: Sensitivity, Stability and parametric analysis

This paper is an updated and edited version of the survey that appeared in Boukari [42]. The original work reported on research since the Fiacco-McCormick 1968 survey [95] that has been undertaken in penalty function methodology and its extensions (exact penalty and augmented Lagrangian); the software that implements penalty and augmented Lagrangian algorithms; and some of the comparative studies that have been carried on during the 1968-88 period The original survey also included some results relative to Karmarker's [I441 method and its extensions. The bulk of the new material here is contained in a new section that deals with recent (roughly, 1988-93) advances in interior-point classical penalty function, exactpenalty, and augmented Lagrangian methods

Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993 ∗

A method for sensitivity analysis in nonlinear programming has recently been developed using the sequential unconstrained minimization technique. It is applied here to perform sensitivity analyses on four example problems to demonstrate the computational feasibility and characteristics of the approach. The sensitivity analysis is conducted along the minimizing trajectory for each problem. The convergence characteristics of the first partial derivatives of the variables and objective function with respect to the parameters in the sensitivity analysis are illustrated.

Anthony V. Fiacco

Papers

Sensitivity analysis for nonlinear programming using penalty methods

Sensitivity and stability analysis for nonlinear programming

Mathematical programming study 21: Sensitivity, Stability and parametric analysis

Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993 ∗

Computational experience in sensitivity analysis for nonlinear programming