  Given multiple non-convex objective functions and objective-specific weights,
Chebyshev scalarization (CS) is a well-known approach to obtain an Exact Pareto
Optimal (EPO), i.e., a solution on the Pareto front (PF) that intersects the
ray defined by the inverse of the weights. First-order optimizers that use the
CS formulation to find EPO solutions encounter practical problems of
oscillations and stagnation that affect convergence. Moreover, when initialized
with a PO solution, they do not guarantee a controlled trajectory that lies
completely on the PF. These shortcomings lead to modeling limitations and
computational inefficiency in multi-task learning (MTL) and multi-criteria
decision-making (MCDM) methods that utilize CS for their underlying non-convex
multi-objective optimization (MOO). To address these shortcomings, we design a
new MOO method, EPO Search. We prove that EPO Search converges to an EPO
solution and empirically illustrate its computational efficiency and robustness
to initialization. When initialized on the PF, EPO Search can trace the PF and
converge to the required EPO solution at a linear rate of convergence. Using
EPO Search we develop new algorithms: PESA-EPO for approximating the PF in a
posteriori MCDM, and GP-EPO for preference elicitation in interactive MCDM;
experiments on benchmark datasets confirm their advantages over competing
alternatives. EPO Search scales linearly with the number of decision variables
which enables its use for training deep networks. Empirical results on real
data from personalized medicine, e-commerce and hydrometeorology demonstrate
the efficacy of EPO Search for deep MTL.
