Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-like Systems

Abstract

We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, whose trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plain multiobjective steepest descent method (Algorithm 1). A further improvement in terms of efficiency is achieved by avoiding the solution of a quadratic subproblem to compute a common step direction for all objective functions, which is usually required in first order methods. Using a different discretization of our inertial gradient-like dynamical system, we obtain an accelerated multiobjective gradient method that does not require the solution of a subproblem in each step (Algorithm 2). While this algorithm does not converge in general, it yields good results on test problems while being faster than standard steepest descent by two to three orders of magnitude.

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“…Further, they proved the convergence rate characterised by the merit function is O(1/k 2 ) just when the extrapolation parameter β k = k−1 k+2 which is the special case of (10); this algorithm has the similar convergence behaviors to the accelerated algorithm given by [15] which has been discussed in [21]. However, the convergence rate of the accelerated proximal gradient algorithms when α > 3 has not been obtained both in the discrete case [21] and the continuous case [22]. This question also was raised as an open question in the Section 9 of [21].…”

“…Remark 1 Recently, it is observed that the fast multiobjective gradient methods with Nesterov acceleration also proposed through the inertial gradient-like dynamical systems [21,22]. In [21], the authors considered the problem (1) with h(x) = 0 and established the inertial first order method for MOP by an discretization of the differential equation.…”

“…Remark 1 Recently, it is observed that the fast multiobjective gradient methods with Nesterov acceleration also proposed through the inertial gradient-like dynamical systems [21,22]. In [21], the authors considered the problem (1) with h(x) = 0 and established the inertial first order method for MOP by an discretization of the differential equation. Further, they proved the convergence rate characterised by the merit function is O(1/k 2 ) just when the extrapolation parameter β k = k−1 k+2 which is the special case of (10); this algorithm has the similar convergence behaviors to the accelerated algorithm given by [15] which has been discussed in [21].…”

“…In [21], the authors considered the problem (1) with h(x) = 0 and established the inertial first order method for MOP by an discretization of the differential equation. Further, they proved the convergence rate characterised by the merit function is O(1/k 2 ) just when the extrapolation parameter β k = k−1 k+2 which is the special case of (10); this algorithm has the similar convergence behaviors to the accelerated algorithm given by [15] which has been discussed in [21]. However, the convergence rate of the accelerated proximal gradient algorithms when α > 3 has not been obtained both in the discrete case [21] and the continuous case [22].…”

“…However, the convergence rate of the accelerated proximal gradient algorithms when α > 3 has not been obtained both in the discrete case [21] and the continuous case [22]. This question also was raised as an open question in the Section 9 of [21]. Fortunately, based on the Lemma 6.3, Lemma 6.5 in [21] and motivated by the Theorem 1, we also can obtain that the convergence rate o(1/k 2 ) also can be obtained when α > 3 in the fast proximal gradient algorithm for MOP [21] which solved their open question [21].…”