arXiv subjects

Vincent Guigues

Publications and source records attributed to Vincent Guigues.

At least 37 records · Page 2Linked to original sources

arXiv2025

Parallelizing the Circumcentered-Reflection Method

This paper introduces the Parallelized Circumcentered Reflection Method (P-CRM), a circumcentric approach that parallelizes the Circumcentered Reflection Method (CRM) for solving Convex Feasibility Problems in affine settings. Beyond feasibility, P-CRM solves the best approximation problem for any finite collection of affine subspaces; that is, it not only finds a feasible point but directly computes the projection of an initial point onto the intersection. Within a fully self-contained scheme, we also introduce the Framework for the Simultaneous Projection Method (F-SPM) which includes Cimmino's method as a special case. Theoretical results show that both P-CRM and F-SPM achieve linear convergence. Moreover, P-CRM converges at a rate that is at least as fast as, and potentially superior to, the best convergence rate of F-SPM. As a byproduct, this also yields a new and simplified convergence proof for Cimmino's method. Numerical experiments show that P-CRM is competitive compared to CRM and indicate that it offers a scalable and flexible alternative, particularly suited for large-scale problems and modern computing environments.

math.OC↗

arXiv2024

Estimation of Spatiotemporal Poisson Processes with Some Missing Location Data

We consider models for spatiotemporal Poisson processes with some missing location data. We discuss four models that make provision for missing location data, and their estimation. The corresponding code is available on GitHub as an extension of LASPATED at https://github.com/vguigues/LASPATED/Missing_Data. We tested our models using the process of emergency call arrivals to an emergency medical service where the emergency reports often omit the location of the emergency. We show the difference made by using models that make provision for missing location data.

stat.ME↗

arXiv2024

Bundle methods with quadratic cuts for deterministic and stochastic strongly convex optimization problems

We introduce two new methods for deterministic convex optimization problems: QCC (Quadratic Cuts for Convex optimization) and QB (Quadratic Bundle method). We prove the complexity of these methods for composite optimization problems which are the sum of a convex function $\tilde h$ and of a strongly convex function $\tilde f$ with parameter $μ$. These methods use as building blocks quadratic approximations of the strongly convex function $\tilde f$ where the quadratic terms are of form $\fracμ{2}\|\cdot-x_i\|^2$ for trial points $x_i$ computed along iterations (when $μ=0$ the building blocks are linear approximations). We extend the idea of using quadratic approximations to pieces of the objective for some multistage stochastic optimization problems which have strongly convex recourse functions that we approximate as a maximum of quadratic cuts. We call DASC (Dynamic Approximation for Strongly Convex optimzation) the corresponding optimization method. When the cuts are linear, the method boils down to the popular Stochastic Dual Dynamic Programming (SDDP) method. We provide conditions ensuring strong convexity of the recourse functions and prove the convergence of DASC. Numerical experiments illustrate the performance and correctness of DASC, with DASC being much quicker than SDDP for large values of the constants of strong convexity.

math.OC↗

arXiv2024

Operation of an ambulance fleet under uncertainty

We introduce two new optimization models for the dispatch of ambulances. The first model, called the ambulance selection problem, is used when an emergency call arrives to decide whether an ambulance should be dispatched for that call, and if so, which ambulance should be dispatched, or whether the request should be put in a queue of waiting requests. The second model, called the ambulance reassignment problem, is used when an ambulance finishes its current task to decide whether the ambulance should be dispatched to a request waiting in queue, and if so, which request, or whether the ambulance should be dispatched to an ambulance staging location, and if so, which ambulance staging location. These decisions affect not only the emergency call and ambulance under consideration, but also the ability of the ambulance fleet to service future calls. There is uncertainty regarding the locations, arrival times, and types of future calls. We propose a rolling horizon approach that combines the current decisions to be made as first-stage decisions with second-stage models that represent the ability of the ambulance fleet to service future calls. The second-stage optimization problems can be formulated as large-scale deterministic integer linear programs. We propose a column generation algorithm to solve the continuous relaxation of these second-stage problems. The optimal objective values of these second-stage continuous relaxations are used to make approximately optimal first-stage decisions. We compare our resulting dispatch policy with popular decision rules for Rio de Janeiro emergency medical service, based on data of more than 2 years of emergency calls for that service. These tests show that our proposed policy results in smaller response times than the popular decision rules.

math.OC↗

arXiv2024

LASPATED: a Library for the Analysis of SPAtio-TEmporal Discrete data

We describe methods, tools, and a software library called LASPATED, available on GitHub (at https://github.com/vguigues/) to fit models using spatio-temporal data and space-time discretization. A video tutorial for this library is available on YouTube. We consider two types of methods to estimate a non-homogeneous Poisson process in space and time. The methods approximate the arrival intensity function of the Poisson process by discretizing space and time, and estimating arrival intensity as a function of subregion and time interval. With such methods, it is typical that the dimension of the estimator is large relative to the amount of data, and therefore the performance of the estimator can be improved by using additional data. The first method uses additional data to add a regularization term to the likelihood function for calibrating the intensity of the Poisson process. The second method uses additional data to estimate arrival intensity as a function of covariates. We describe a Python package to perform various types of space and time discretization. We also describe two packages for the calibration of the models, one in Matlab and one in C++. We demonstrate the advantages of our methods compared to basic maximum likelihood estimation with simulated and real data. The experiments with real data calibrate models of the arrival process of emergencies to be handled by the Rio de Janeiro emergency medical service.

stat.ME↗

arXiv2023

A single cut proximal bundle method for stochastic convex composite optimization

This paper considers optimization problems where the objective is the sum of a function given by an expectation and a closed convex composite function, and proposes stochastic composite proximal bundle (SCPB) methods for solving it. Complexity guarantees are established for them without requiring knowledge of parameters associated with the problem instance. Moreover, it is shown that they have optimal complexity when these problem parameters are known. To the best of our knowledge, this is the first proximal bundle method for stochastic programming able to deal with continuous distributions. Finally, we present computational results showing that SCPB substantially outperforms the robust stochastic approximation (RSA) method in all instances considered.

math.OC↗

arXiv2023

Risk-Averse Stochastic Optimal Control: an efficiently computable statistical upper bound

In this paper, we discuss an application of the SDDP type algorithm to nested risk-averse formulations of Stochastic Optimal Control (SOC) problems. We propose a construction of a statistical upper bound for the optimal value of risk-averse SOC problems. This outlines an approach to a solution of a long standing problem in that area of research. The bound holds for a large class of convex and monotone conditional risk mappings. Finally, we show the validity of the statistical upper bound to solve a real-life stochastic hydro-thermal planning problem.

math.OC↗

arXiv2021

On the strong concavity of the dual function of an optimization problem

We provide three new proofs of the strong concavity of the dual function of some convex optimization problems. For problems with nonlinear constraints, we show that the the assumption of strong convexity of the objective cannot be weakened to convexity and that the assumption that the gradients of all constraints at the optimal solution are linearly independent cannot be further weakened. Finally, we illustrate our results with several examples.

math.OC↗

arXiv2021

Stochastic Dynamic Cutting Plane for multistage stochastic convex programs

We introduce StoDCuP (Stochastic Dynamic Cutting Plane), an extension of the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve multistage stochastic convex optimization problems. At each iteration, the algorithm builds lower affine functions not only for the cost-to-go functions, as SDDP does, but also for some or all nonlinear cost and constraint functions. We show the almost sure convergence of StoDCuP. We also introduce an inexact variant of StoDCuP where all subproblems are solved approximately (with bounded errors) and show the almost sure convergence of this variant for vanishing errors.

math.OC↗

arXiv2021

Inexact cuts in SDDP applied to multistage stochastic nondifferentiable problems

In [13], an Inexact variant of Stochastic Dual Dynamic Programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the method. That variant of SDDP was studied in [13] for linear and for differentiable nonlinear Multistage Stochastic Programs (MSPs). In this paper, we extend ISDDP to nondifferentiable MSPs. We first provide formulas for inexact cuts for value functions of convex nondifferentiable optimization problems. We then combine these cuts with SDDP to describe ISDDP for nondifferentiable MSPs and analyze the convergence of the method. More precisely, for a problem with T stages, we show that for errors bounded from above by epsilon, the limit superior and limit inferior of sequences of upper and lower bounds on the optimal value of the problem are at most at distance 3*epsilon*T to the optimal value and that for asymptotically vanishing errors ISDDP converges to an optimal policy. [13] V. Guigues, Inexact cuts in Stochastic Dual Dynamic Programming, Siam Journal on Optimization, 30(1), 407-438, 2020.

math.OC↗

arXiv2021

Constant Depth Decision Rules for multistage optimization under uncertainty

In this paper, we introduce a new class of decision rules, referred to as Constant Depth Decision Rules (CDDRs), for multistage optimization under linear constraints with uncertainty-affected right-hand sides. We consider two uncertainty classes: discrete uncertainties which can take at each stage at most a fixed number d of different values, and polytopic uncertainties which, at each stage, are elements of a convex hull of at most d points. Given the depth mu of the decision rule, the decision at stage t is expressed as the sum of t functions of mu consecutive values of the underlying uncertain parameters. These functions are arbitrary in the case of discrete uncertainties and are poly-affine in the case of polytopic uncertainties. For these uncertainty classes, we show that when the uncertain right-hand sides of the constraints of the multistage problem are of the same additive structure as the decision rules, these constraints can be reformulated as a system of linear inequality constraints where the numbers of variables and constraints is O(1)(n+m)d^mu N^2 with n the maximal dimension of control variables, m the maximal number of inequality constraints at each stage, and N the number of stages. As an illustration, we discuss an application of the proposed approach to a Multistage Stochastic Program arising in the problem of hydro-thermal production planning with interstage dependent inflows. For problems with a small number of stages, we present the results of a numerical study in which optimal CDDRs show similar performance, in terms of optimization objective, to that of Stochastic Dual Dynamic Programming (SDDP) policies, often at much smaller computational cost.

math.OC↗

arXiv2020

Duality and sensitivity analysis of multistage linear stochastic programs

We investigate the dual of a Multistage Stochastic Linear Program (MSLP) to study two questions for this class of problems. The first of these questions is the study of the optimal value of the problem as a function of the involved parameters. For this sensitivity analysis problem, we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them. In this context, the second question we address is the study of solution methods for the dual problem. Writing Dynamic Programming (DP) equations for the dual, we can use an SDDP type method, called Dual SDDP, which solves these DP equations. However, applying this method will only be possible if the Relatively Complete Recourse (RCR) holds for the dual. Since the RCR assumption may fail to hold, we design two variants of Dual SDDP, namely Dual SDDP with penalizations and Dual SDDP with feasibility cuts. The study of this second question allows us to take a fresh look at the notoriously difficult to solve class of MSLP with interstage dependent cost coefficients. Indeed, for this class of problems, cost-to-go functions are non-convex and solution methods were so far using SDDP for a Markov chain approximation of the cost coefficients process. For these problems, we propose to apply Dual SDDP with penalizations to the cost-to-go functions of the dual which are concave. This algorithm converges to the optimal value of the problem. Finally, as a proof of concept of the tools developed, we present the results of numerical experiments computing the sensitivity of the optimal value of an inventory problem as a function of parameters of the demand process and compare Primal and Dual SDDP on the inventory and a hydro-thermal planning problems.

math.OC↗

arXiv2020

Inexact Stochastic Mirror Descent for two-stage nonlinear stochastic programs

We introduce an inexact variant of Stochastic Mirror Descent (SMD), called Inexact Stochastic Mirror Descent (ISMD), to solve nonlinear two-stage stochastic programs where the second stage problem has linear and nonlinear coupling constraints and a nonlinear objective function which depends on both first and second stage decisions. Given a candidate first stage solution and a realization of the second stage random vector, each iteration of ISMD combines a stochastic subgradient descent using a prox-mapping with the computation of approximate (instead of exact for SMD) primal and dual second stage solutions. We propose two variants of ISMD and show the convergence of these variants to the optimal value of the stochastic program. We show in particular that under some assumptions, ISMD has the same convergence rate as SMD. The first variant of ISMD and its convergence analysis are based on the formulas for inexact cuts of value functions of convex optimization problems shown recently in [4]. The second variant of ISMD and the corresponding convergence analysis rely on new formulas that we derive for inexact cuts of value functions of convex optimization problems assuming that the dual function of the second stage problem for all fixed first stage solution and realization of the second stage random vector, is strongly concave. We show that this assumption of strong concavity is satisfied for some classes of problems and present the results of numerical experiments on two simple two-stage problems which show that solving approximately the second stage problem for the first iterations of ISMD can help us obtain a good approximate first stage solution quicker than with SMD. [4] V. Guigues, Inexact decomposition methods for solving deterministic and stochastic convex dynamic programming equations, arXiv, available at arXiv:1707.00812, 2017.

math.OC↗

arXiv2020

Regularized Stochastic Dual Dynamic Programming for convex nonlinear optimization problems

We define a regularized variant of the Dual Dynamic Programming algorithm called REDDP (REgularized Dual Dynamic Programming) to solve nonlinear dynamic programming equations. We extend the algorithm to solve nonlinear stochastic dynamic programming equations. The corresponding algorithm, called SDDP-REG, can be seen as an extension of a regularization of the Stochastic Dual Dynamic Programming (SDDP) algorithm recently introduced which was studied for linear problems only and with less general prox-centers. We show the convergence of REDDP and SDDP-REG. We assess the performance of REDDP and SDDP-REG on portfolio models with direct transaction and market impact costs. In particular, we propose a risk-neutral portfolio selection model which can be cast as a multistage stochastic second-order cone program. The formulation is motivated by the impact of market impact costs on large portfolio rebalancing operations. Numerical simulations show that REDDP is much quicker than DDP on all problem instances considered (up to 184 times quicker than DDP) and that SDDP-REG is quicker on the instances of portfolio selection problems with market impact costs tested and much faster on the instance of risk-neutral multistage stochastic linear program implemented (8.2 times faster).

math.OC↗

arXiv2019

Multicut decomposition methods with cut selection for multistage stochastic programs

math.OC↗

arXiv2019

Single cut and multicut SDDP with cut selection for multistage stochastic linear programs: convergence proof and numerical experiments

We introduce a variant of Multicut Decomposition Algorithms (MuDA), called CuSMuDA (Cut Selection for Multicut Decomposition Algorithms), for solving multistage stochastic linear programs that incorporates a class of cut selection strategies to choose the most relevant cuts of the approximate recourse functions. This class contains Level 1 and Limited Memory Level 1 cut selection strategies, initially introduced for respectively Stochastic Dual Dynamic Programming (SDDP) and Dual Dynamic Programming (DDP). We prove the almost sure convergence of the method in a finite number of iterations and obtain as a by-product the almost sure convergence in a finite number of iterations of SDDP combined with our class of cut selection strategies. We compare the performance of MuDA, SDDP, and their variants with cut selection (using Level 1 and Limited Memory Level 1) on several instances of a portfolio problem and of an inventory problem. On these experiments, in general, SDDP is quicker (i.e., satisfies the stopping criterion quicker) than MuDA and cut selection allows us to decrease the computational bulk with Limited Memory Level 1 being more efficient (sometimes much more) than Level 1.

math.OC↗

arXiv2019

Multistage stochastic programs with a random number of stages: dynamic programming equations, solution methods, and application to portfolio selection

We introduce the class of multistage stochastic optimization problems with a random number of stages. For such problems, we show how to write dynamic programming equations and detail the Stochastic Dual Dynamic Programming algorithm to solve these equations. Finally, we consider a portfolio selection problem over an optimization period of random duration. For several instances of this problem, we show the gain obtained using a policy that takes the random duration of the number of stages into account over a policy built taking a fixed number of stages (namely the maximal possible number of stages).

math.OC↗

arXiv2019

Inexact cuts in Deterministic and Stochastic Dual Dynamic Programming applied to linear optimization problems

We introduce an extension of Dual Dynamic Programming (DDP) to solve linear dynamic programming equations. We call this extension IDDP-LP which applies to situations where some or all primal and dual subproblems to be solved along the iterations of the method are solved with a bounded error (inexactly). We provide convergence theorems both in the case when errors are bounded and for asymptotically vanishing errors. We extend the analysis to stochastic linear dynamic programming equations, introducing Inexact Stochastic Dual Dynamic Programming for linear programs (ISDDP-LP), an inexact variant of SDDP applied to linear programs corresponding to the situation where some or all problems to be solved in the forward and backward passes of SDDP are solved approximately. We also provide convergence theorems for ISDDP-LP for bounded and asymptotically vanishing errors. Finally, we present the results of numerical experiments comparing SDDP and ISSDP-LP on a portfolio problem with direct transation costs modelled as a multistage stochastic linear optimization problem. On these experiments, ISDDP-LP allows us to obtain a good policy faster than SDDP.

math.OC↗