SCRO / Journées de l'optimisation

HEC Montréal, 29-31 mai 2023


HEC Montréal, 29 — 31 mai 2023

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NLS Non-Linear Systems

31 mai 2023 15h40 – 17h20

Salle: Transat (jaune)

Présidée par Jean-Luc Lupien

4 présentations

  • 15h40 - 16h05

    On spline regression using interior-point trust-region optimization

    • Theodore Stanoev, prés., Memorial University of Newfoundland

    We present an approach to nonlinear regression using the natural cubic spline. In this approach, we use an interior-point trust-region optimization method, subject to bounds, to adjust the values of the knot coordinates to minimize the residual sum of squares with a given dataset. To do so, we parametrize the spline coefficients as a function of the knots, from which we build the quadratic model for the trust region. As the knot abscissae must be strictly increasing throughout the spline domain, we require an interior-point approach to maintain model feasibility during the optimization. With a multistart procedure, we obtain solutions that satisfy second-order optimality conditions, given different initial values and an increasing number of spline segments. We perform a posteriori statistical analysis using the Bayesian Information Criterion to determine the best model from the solution set. This procedure yields a spline regression with unevenly spaced knots and automatically determined model dimensionality, with practical applications to the curve fitting and noise reduction of data.

  • 16h05 - 16h30

    Single loop Lagrangian-based method for nonconvex optimization with nonlinear constraints

    • Dimitri Papadimitriou,
    • Dimitri Papadimitriou, prés., 3nLab, BeRC

    The augmented Lagrangian-based (ALM) method is one of the most common approaches for solving linearly and nonlinearly constrained problems. However, for nonconvex objectives, the handling of nonlinear inequality constraints remains challenging. We propose a single loop ALM-based algorithmic scheme with Backtracking Line Search to solve nonconvex optimization problems including both nonlinear equality and inequality constraints. Moreover, when some variables belong to the real Hilbert space and others to the integer space, we obtain a computational method for solving nonconvex mixed-integer/-binary nonlinear problems for which the nonconvexity is not limited to the integrality constraints. Further, we demonstrate the local convergence properties as well as the iteration complexity (i.e., the number of iterations required to obtain an approximate KKT point) of the proposed algorithmic scheme. The performances of the proposed algorithm are then numerically evaluated to solve a multi-constrained network design problem. For this purpose, extensive numerical executions on a set of instances extracted from the SNDlib repository are performed to analyze its behavior and performance as well as identify potential improvements.

  • 16h30 - 16h55

    CANCELED : Practical introduction to nonlinear programming with Artelys Knitro

    • Renaud Saltet, prés., Artelys

    Problems encountered by operational research experts are generally complex and intrinsically nonlinear. It is therefore necessary to be familiar with nonlinear optimization methods using the appropriate tools.

    This tutorial will present some examples of nonlinear problems, usually solved using a linear relaxation, for which a direct approach is more relevant.

    This tutorial is for an audience familiar with combinatorial programming and linear programming. The participants will practice nonlinear programming methods using the Artelys Knitro solver in a Python notebook, through different examples (

    The goal of this tutorial is to enable participants, when confronted with a nonlinear problem, to identify what type of nonlinear problem it is, what methods and tools are adapted to solve it, and how to use them.

    Notes: This session presents practical exercises to be performed by the participants. A computer and an Internet connection are required to participate.

  • 16h55 - 17h20

    An Online Newton's Method for Time-varying Equality Constraints

    • Jean-Luc Lupien, prés., Polytechnique Montréal
    • Antoine Lesage-Landry, Polytechnique Montréal

    We consider online optimization problems with time-varying linear equality constraints. In this framework, an agent makes sequential decisions using only prior information. At every round, the agent suffers an environment-determined loss and must satisfy time-varying constraints. Both the loss functions and the constraints can be chosen adversarially. We propose the Online Projected Equality-constrained Newton Method (OPEN-M) to tackle this family of problems. We obtain sublinear dynamic regret and constraint violation bounds for OPEN-M under mild conditions. Namely, smoothness of the loss function and boundedness of the inverse Hessian at the optimum are required, but not convexity. Finally, we show OPEN-M outperforms state-of-the-art online constrained optimization algorithms in a numerical network flow application.