Using Interior-Point Methods within an Outer Approximation Framework for Mixed Integer Nonlinear Programming

Benson, Hande Y.

doi:10.1007/978-1-4614-1927-3_7

Cited by 12 publications

(7 citation statements)

References 29 publications

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“…The source code of MILANO is available on the project website. The main focus of this code is to develop efficient warm-starting methods for interior-point methods (Benson 2011(Benson , 2012) so as to make them more effective at solving MINLPs.…”

Section: Convex Minlp Solversmentioning

confidence: 99%

Mixed-integer nonlinear optimization

et al. 2013

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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. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

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Section: Convex Minlp Solversmentioning

confidence: 99%

Mixed-integer nonlinear optimization

et al. 2013

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“…A ratio reformulation was used to smooth the underlying SOCPs. The primal-dual penalty interior-point method, modified from that presented in [10,11], was then used to provide warmstarts, regularization, and infeasibility detection capabilities, and the modification also exploited the structure of the MISOCP. We have implemented both branchand-bound and outer approximation frameworks that use this method and use them to solve portfolio optimization problems.…”

Section: Resultsmentioning

confidence: 99%

“…Outer approximation alternates between the solution of an NLP obtained by fixing the integer variables and of an MILP obtained using linearizations of the objective function and the constraints at the solutions of the NLP. These methods and their use in conjunction with the primal-dual penalty interior-point method were analyzed in [10,11]. We refer the reader to these papers for further details.…”

Section: Handling the Discrete Variablesmentioning

confidence: 99%

“…In contrast, starting an interior-point method from the optimal solution of another problem causes issues due to at least one of a complementary pair of primal-dual variables already being at its bound. A thorough analysis of the numerical difficulties is presented in [13,14] for general linear and nonlinear programming warmstarts, respectively, and in [10,11] for warmstarts within branchand-bound and outer approximation frameworks, respectively, for mixed-integer nonlinear programming. In all instances, it is shown that a standard interior-point method, applied directly to the original problem, will not only fail to warmstart but fatally stall if initialized from the optimal solution of a previously solved problem.…”

Section: Warmstartingmentioning

confidence: 99%

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Smoothing and Regularization for Mixed-Integer Second-Order Cone Programming with Applications in Portfolio Optimization

Benson

Sağlam

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Second-order cone programming problems (SOCPs) have been well studied in literature, and computationally efficient implementations of solution algorithms exist. In this paper, we study an extension: mixed-integer second-order cone programming problems (MISOCPs). Our focus is on designing an algorithm for solving the underlying SOCPs as nonlinear programming problems (NLPs) within two existing frameworks, branch-and-bound and outer approximation, for mixed-integer nonlinear programming (MINLP). We pay particular attention to resolving the nondifferentiability of the underlying SOCPs via a smoothing reformulation, as well as warmstarting and infeasibility detection using a regularization. We investigate the application of our proposed techniques to portfolio optimization problems that can be formulated as MISOCPs, and preliminary numerical results using the Matlab-based optimization package MILANO (Benson, MILANO-a Matlab-based code for mixed-integer linear and nonlinear optimization. http://www. pages.drexel.edu/~hvb22/milano) are provided.

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“…The feasibility criteria for trajectories require that the robots' kinematic and dynamic constraints be satisfied, along with the imperatives of avoiding collisions and obeying the communication connectivity constraints. The receding horizon formulation is implemented in MAT-LAB, which is interfaced with the MINLP solver MILANO [5]. MILANO is a MATLAB-based solver for mixed-integer linear and nonlinear programming problems.…”

Section: Introductionmentioning

confidence: 99%

Decentralized multi-vehicle path coordination under communication constraints

Abichandani¹,

Benson²,

Kam³

2011

2011 IEEE/RSJ International Conference on Intelligent Robots and Systems

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Abstract-We present a mathematical programming based decentralized framework to generate time optimal velocity profiles for a group of path constrained mobile vehicle robots subject to communication connectivity constraints. Each vehicle robot starts from a fixed start point and moves towards a goal point along a fixed path so as to avoid collisions with other robots, and remain in communication connectivity with other robots. The main contribution of this paper is the discrete time decentralized Receding Horizon Mixed Integer Nonlinear Programming (RH-MINLP) formulation of the multi-vehicle path coordination problem with constraints on kinematics, dynamics, collision avoidance, and communication connectivity, and the application of state-of-the-art MINLP solution techniques. We test scenarios involving up to ten (10) robots to demonstrate (i) the effect of communication connectivity requirements on robot velocity profiles; and (ii) the dependence of the solution computation time on communication connectivity requirements.

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Using Interior-Point Methods within an Outer Approximation Framework for Mixed Integer Nonlinear Programming

Cited by 12 publications

References 29 publications

Mixed-integer nonlinear optimization

Mixed-integer nonlinear optimization

Smoothing and Regularization for Mixed-Integer Second-Order Cone Programming with Applications in Portfolio Optimization

Decentralized multi-vehicle path coordination under communication constraints

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