We propose a criterion space search algorithm for bi-objective mixed integer linear programming problems. The Pareto frontier of these problems can have a complex structure, as it can include isolated points, open, half-open and closed line segments. Therefore, its exact detection is a hard computational task. Our algorithm handles this issue by alternating the resolution of single objective mixed integer linear problems with bi-objective linear ones. The performance of the algorithm is improved using suitably defined cuts and related strategies. Under specific assumptions, we can prove that the exact Pareto frontier can be detected in a finite number of iterations. Experimental results on a test-bed of instances and a comparison with available solvers is presented, showing the notably good performance of our approach in terms of accuracy of the Pareto frontier detected.

A standard tool for modelling real-world optimisation problems is mixed-integer programming (MIP). However, for many of these problems there is either incomplete information describing variable relations, or the relations between variables are highly complex. To overcome both these hurdles, machine learning (ML) models are often used and embedded in the MIP as surrogate models to represent these relations. Due to the large amount of available ML frameworks, formulating ML models into MIPs is highly non-trivial. We present PySCIPOpt-ML, a tool for the automatic MIP formulation of trained ML models, allowing easy integration of ML constraints into MIPs. In this presentation we will discuss the MIP formulations of the embedded ML models, and present computational results on the scale of ML models that can easily be embedded. In addition, we introduce a library of MIP instances with embedded ML constraints.

We present QUBO.jl, an end-to-end Julia package for working with QUBO (Quadratic Unconstrained Binary Optimization) instances. This tool aims to convert a broad range of JuMP problems for straightforward application in many physics and physics-inspired solution methods whose standard optimization form is equivalent to the QUBO. These methods include quantum annealing, quantum gate-circuit optimization algorithms (Quantum Optimization Alternating Ansatz, Variational Quantum Eigensolver), other hardware-accelerated platforms, such as Coherent Ising Machines and Simulated Bifurcation Machines, and more traditional methods such as simulated annealing.
Besides working with reformulations, QUBO.jl allows its users to interface with the aforementioned hardware, sending QUBO models in various file formats and retrieving results for subsequent analysis. QUBO.jl was written as a JuMP / MathOptInterface (MOI) layer that automatically maps between the input and output frames, thus providing a smooth modeling experience. As a result of the infrastructure developed to work with QUBO formulations and solvers, we present QUBOLib, a software library and instance collection designed to enable the continuous benchmarking of QUBO solvers as they evolve.