A trajectory-based method for mixed integer nonlinear programming problems (original) (raw)

A Trajectory-Based Method for Constrained Nonlinear Optimization Problems

Journal of Optimization Theory and Applications, 2018

A trajectory-based method for solving constrained nonlinear optimization problems is proposed. The method is an extension of a trajectory-based method for unconstrained optimization. The optimization problem is transformed into a system of second-order differential equations with the aid of the augmented Lagrangian. Several novel contributions are made, including a new penalty parameter updating strategy, an adaptive step size routine for numerical integration and a scaling mechanism. A new criterion is suggested for the adjustment of the penalty parameter. Global convergence properties of the method are established.

An Exact Penalty Approach for Mixed Integer Nonlinear Programming Problems

American Journal of Operations Research, 2011

We propose an exact penalty approach for solving mixed integer nonlinear programming (MINLP) problems by converting a general MINLP problem to a finite sequence of nonlinear programming (NLP) problems with only continuous variables. We express conditions of exactness for MINLP problems and show how the exact penalty approach can be extended to constrained problems.

Multiphase Mixed-Integer Optimal Control Approach to Aircraft Trajectory Optimization

Journal of Guidance, Control, and Dynamics, 2013

In this paper, an approach to aircraft trajectory optimization is presented in which integer and continuous variables are considered. Integer variables model decision-making processes, and continuous variables describe the state of the aircraft, which evolves according to differential-algebraic equations. The problem is formulated as a multiphase mixed-integer optimal control problem. It is transcribed into a mixed-integer nonlinear programming problem by applying a fifth degree Gauss-Lobatto direct collocation method and is then solved using a nonlinearprogramming-based branch-and-bound algorithm. The approach is applied to the following en route flight planning problem: Given an aircraft point mass model, a wind forecast, an airspace structure, and the relevant flying information regions with their associated overflying costs, find the control inputs that steer the aircraft from the initial fix to the final fix, following a route of waypoints while minimizing the fuel consumption and overflying costs during the flight. The decision-making process arises in determining the optimal sequence of waypoints. The optimal times at which the waypoints are to be overflown are also to be determined. Numerical results are presented and discussed, showing the effectiveness of the approach.

New algorithms for mixed-integer dynamic optimization

Computers & Chemical Engineering, 2003

Mixed-integer dynamic optimization (MIDO) problems arise in chemical engineering whenever discrete and continuous decisions are to be made for a system described by a transient model. Areas of application include integrated design and control, synthesis of reactor networks, reduction of kinetic mechanisms and optimization of hybrid systems. This article presents new formulations and algorithms for solving MIDO problems. The algorithms are based on decomposition into primal, dynamic optimization and master, mixed-integer linear programming sub-problems. They do not depend on the use of a particular primal dynamic optimization method and they do not require the solution of an intermediate adjoint problem for constructing the master problem, even when the integer variables appear explicitly in the differential Á/algebraic equation system. The practical potential of the algorithms is demonstrated with two distillation design and control optimization examples. #

Local Search for Mixed-Integer Nonlinear Optimization: A Methodology and an Application

Lecture Notes in Computer Science, 2011

A methodology is presented for tackling mixed-integer nonlinear optimization problems by local search, in particular large-scale real-life problems. This methodology is illustrated through the localsearch heuristic implemented for solving an energy management problem posed by the EDF company in the context of the ROADEF/EURO Challenge 2010, an international competition of applied optimization. Our local-search approach is pure and direct: the problem is tackled frontally, without decomposition nor hybridization. In this way, both combinatorial and continuous decisions can be modified by a move during the search. Then, our work focuses on the diversification by the moves and on the performance of the incremental evaluation machinery. Exploring millions of feasible solutions within one hour of running time, the resulting local search allows us to obtain among the best results of the competition, in which 44 teams from 25 countries were engaged.

Global optimization of mixed-integer nonlinear problems

AIChE Journal, 2000

Two novel deterministic global optimization algorithms for nonconvex mixed-integer problems (MINLPs) are proposed, using the advances of the BB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure Mixed-Integer BB algorithm (SMIN-BB addresses problems with nonconvexities in the continuous variables and linear and mixed-bilinear participation of the binary variables. The General Structure Mixed-Integer BB algorithm (GMIN-BB), is applicable to a very general class of problems for which the continuous relaxation is twice continuously di erentiable. Both algorithms are developed using the concepts of branch-and-bound, but they di er in their approach to each of the required steps. The SMIN-BB algorithm is based on the convex underestimation of the continuous functions while the GMIN-BB algorithm is centered around the convex relaxation of the entire problem. Both algorithms rely on optimization or interval based variable bound updates to enhance e ciency. A series of medium-size engineering applications demonstrates the performance of the algorithms. Finally, a comparison of the two algorithms on the same problems highlights the value of algorithms which can handle binary or integer variables without reformulation.

Mixed-integer nonlinear optimization

Acta Numerica, 2013

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. T...

Mixed-integer programming for control

American Control Conference, 2005. …, 2005

The article describes how mixed-integer programming (MIP) can be employed for feedback control. MIP can be used to find optimal trajectories subject to integer constraints, which can encode discrete decisions or nonconvexity, for example. This optimization can be performed online within model predictive control (MPC) to implement a feedback control law. The article discusses how to model systems using MIP, the implementation as a MPC, and techniques for fast solutions of the optimizations to make them suitable for real-time use.

Review of nonlinear mixed-integer and disjunctive programming techniques

Optimization and Engineering, 2002

This paper has as a major objective to present a unified overview and derivation of mixed-integer nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.

Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Methods

Chemie Ingenieur Technik, 2014

This work presents a review of the applications of mixed-integer nonlinear programming (MINLP) in process systems engineering (PSE). A review on the main deterministic MINLP solution methods is presented, including an overview of the main MINLP solvers. Generalized disjunctive programming (GDP) is an alternative higher-level representation of MINLP problems. This work reviews some methods for solving GDP models, and techniques for improving MINLP methods through GDP. The paper also provides a high-level review of the applications of MINLP in PSE, particularly in process synthesis, planning and scheduling, process control and molecular computing.