6 research outputs found
Constraint qualification failure in action
This note presents a theoretical analysis of disjunctive constraints featuring unbounded variables. In this framework, classical modeling techniques, including big-M approaches, are not applicable. We introduce a lifted second-order cone formulation of such on/off constraints and discuss related constraint qualification issues. A solution is proposed to avoid solvers' failure.Financial support by grants: Digiteo Emergence ‘‘PASO’’, Digiteo
Chair 2009-14D ‘‘RMNCCO’’, Digiteo Emergence 2009-55D ‘‘ARM’’
is gratefully acknowledged
Mixed-integer convex representability
Motivated by recent advances in solution methods for mixed-integer convex
optimization (MICP), we study the fundamental and open question of which sets
can be represented exactly as feasible regions of MICP problems. We establish
several results in this direction, including the first complete
characterization for the mixed-binary case and a simple necessary condition for
the general case. We use the latter to derive the first non-representability
results for various non-convex sets such as the set of rank-1 matrices and the
set of prime numbers. Finally, in correspondence with the seminal work on
mixed-integer linear representability by Jeroslow and Lowe, we study the
representability question under rationality assumptions. Under these
rationality assumptions, we establish that representable sets obey strong
regularity properties such as periodicity, and we provide a complete
characterization of representable subsets of the natural numbers and of
representable compact sets. Interestingly, in the case of subsets of natural
numbers, our results provide a clear separation between the mathematical
modeling power of mixed-integer linear and mixed-integer convex optimization.
In the case of compact sets, our results imply that using unbounded integer
variables is necessary only for modeling unbounded sets
Double Duality: Variational Primal-Dual Policy Optimization for Constrained Reinforcement Learning
We study the Constrained Convex Markov Decision Process (MDP), where the goal
is to minimize a convex functional of the visitation measure, subject to a
convex constraint. Designing algorithms for a constrained convex MDP faces
several challenges, including (1) handling the large state space, (2) managing
the exploration/exploitation tradeoff, and (3) solving the constrained
optimization where the objective and the constraint are both nonlinear
functions of the visitation measure. In this work, we present a model-based
algorithm, Variational Primal-Dual Policy Optimization (VPDPO), in which
Lagrangian and Fenchel duality are implemented to reformulate the original
constrained problem into an unconstrained primal-dual optimization. Moreover,
the primal variables are updated by model-based value iteration following the
principle of Optimism in the Face of Uncertainty (OFU), while the dual
variables are updated by gradient ascent. Moreover, by embedding the visitation
measure into a finite-dimensional space, we can handle large state spaces by
incorporating function approximation. Two notable examples are (1) Kernelized
Nonlinear Regulators and (2) Low-rank MDPs. We prove that with an optimistic
planning oracle, our algorithm achieves sublinear regret and constraint
violation in both cases and can attain the globally optimal policy of the
original constrained problem
Branching strategies for mixed-integer programs containing logical constraints and decomposable structure
Decision-making optimisation problems can include discrete selections, e.g. selecting a route, arranging non-overlapping items or designing a network of items. Branch-and-bound (B&B), a widely applied divide-and-conquer framework, often solves such problems by considering a continuous approximation, e.g. replacing discrete variable domains by a continuous superset. Such approximations weaken the logical relations, e.g. for discrete variables corresponding to Boolean variables. Branching in B&B reintroduces logical relations by dividing the search space. This thesis studies designing B&B branching strategies, i.e. how to divide the search space, for optimisation problems that contain both a logical and a continuous structure.
We begin our study with a large-scale, industrially-relevant optimisation problem where the objective consists of machine-learnt gradient-boosted trees (GBTs) and convex penalty functions. GBT functions contain if-then queries which introduces a logical structure to this problem. We propose decomposition-based rigorous bounding strategies and an iterative heuristic that can be embedded into a B&B algorithm. We approach branching with two strategies: a pseudocost initialisation and strong branching that target the structure of GBT and convex penalty aspects of the optimisation objective, respectively. Computational tests show that our B&B approach outperforms state-of-the-art solvers in deriving rigorous bounds on optimality.
Our second project investigates how satisfiability modulo theories (SMT) derived unsatisfiable cores may be utilised in a B&B context. Unsatisfiable cores are subsets of constraints that explain an infeasible result. We study two-dimensional bin packing (2BP) and develop a B&B algorithm that branches on SMT unsatisfiable cores. We use the unsatisfiable cores to derive cuts that break 2BP symmetries. Computational results show that our B&B algorithm solves 20% more instances when compared with commercial solvers on the tested instances.
Finally, we study convex generalized disjunctive programming (GDP), a framework that supports logical variables and operators. Convex GDP includes disjunctions of mathematical constraints, which motivate branching by partitioning the disjunctions. We investigate separation by branching, i.e. eliminating solutions that prevent rigorous bound improvement, and propose a greedy algorithm for building the branches. We propose three scoring methods for selecting the next branching disjunction. We also analyse how to leverage infeasibility to expedite the B&B search. Computational results show that our scoring methods can reduce the number of explored B&B nodes by an order of magnitude when compared with scoring methods proposed in literature. Our infeasibility analysis further reduces the number of explored nodes.Open Acces
Constraint Qualification Failure in Action
Abstract This note presents a theoretical analysis of disjunctive constraints featuring unbounded variables. In this framework, classical modeling techniques, including big-M approaches, are not applicable. We introduce a lifted second-order cone formulation of such on/off constraints and discuss related constraint qualification issues. A solution is proposed to avoid solvers' failure