SCRO / Journées de l'optimisation
HEC Montréal, 2931 mai 2023
CORSJOPT2023
HEC Montréal, 29 — 31 mai 2023
NLS NonLinear Systems
31 mai 2023 15h40 – 17h20
Salle: Transat (jaune)
Présidée par JeanLuc Lupien
4 présentations

15h40  16h05
On spline regression using interiorpoint trustregion optimization
We present an approach to nonlinear regression using the natural cubic spline. In this approach, we use an interiorpoint trustregion 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 interiorpoint approach to maintain model feasibility during the optimization. With a multistart procedure, we obtain solutions that satisfy secondorder 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 Lagrangianbased method for nonconvex optimization with nonlinear constraints
The augmented Lagrangianbased (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 ALMbased 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 mixedinteger/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 multiconstrained 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
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 (https://www.artelys.com/fr/solveurs/knitro/).
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 Timevarying Equality Constraints
We consider online optimization problems with timevarying linear equality constraints. In this framework, an agent makes sequential decisions using only prior information. At every round, the agent suffers an environmentdetermined loss and must satisfy timevarying constraints. Both the loss functions and the constraints can be chosen adversarially. We propose the Online Projected Equalityconstrained Newton Method (OPENM) to tackle this family of problems. We obtain sublinear dynamic regret and constraint violation bounds for OPENM 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 OPENM outperforms stateoftheart online constrained optimization algorithms in a numerical network flow application.