Distributed Convex Optimization Benchmark for Dual Decomposition

Benchmark Dataset: Distributed Convex Programs (Conv) Overview This dataset contains 7,400 randomly generated distributed convex program instances for benchmarking algorithms for distributed convex optimization via dual decomposition. The benchmark was originally introduced by Yfantis et al. (2023). This dataset was generated using the same procedure with a larger number of instances. Problem Formulation Each instance is a constraint-coupled convex program of the form: min sum_{i=1}^{N_s} f_i(x_i) x_1,...,x_{N_s} s.t. sum_{i=1}^{N_s} A_i x_i = 0 x_i^T G_i x_i <= p_i^2, for all i x_i^lb <= x_i <= x_i^ub, for all i where: N_s is the number of subsystems f_i is a convex local objective function, with type drawn uniformly at random per subsystem A_i is the coupling matrix of subsystem `i` G_i is a positive-definite matrix defining the ellipsoidal constraint p_i is the ellipsoid radius Since x_i = 0 is a strictly feasible point for all instances, Slater's condition is satisfied and strong duality holds throughout the benchmark. Local Objective Functions The objective function type for each subsystem is drawn uniformly at random from the following six convex function classes: Name f_i(x_i) Bounds / Parameters Affine c_i^T x_i + a_i x_i in [-10, 10], a_i, [c_i]_l ~ N(0,1) Quadratic (1/2) x_i^T H_i x_i + c_i^T x_i x_i in [-10, 10], H_i constructed as N_i^T N_i with N_i ~ N(0,1) Powers sum_l ( [x_i]_l + [c_i]_l )^{[a_i]_l} ower bound derived from min(c_i), [a_i]_l, [c_i]_l ~ U_c(1,5) Exponential sum_l exp( [c_i]_l * [x_i]_l + [a_i]_l ) x_i in [-10, 10], [a_i]_l, [c_i]_l ~ N(0,1) Negative log -sum_l [c_i]_l * log( [x_i]_l + [a_i]_l ) lower bound derived from min(c_i), [a_i]_l, [c_i]_l ~ U_c(1,5) Negative entropy sum_l [a_i]_l * [x_i]_l * log( [x_i]_l + [b_i]_l ) lower bound derived from min(c_i), [a_i]_l, [c_i]_l ~ U_c(1,5) Parameter Generation Parameter Distribution / Method Coupling matrix A_i Element-wise product B_i ∘ C_i, where B_i is drawn from a continuous uniform distribution and C_i from a discrete uniform distribution Ellipsoid matrix G_i Constructed as N_i^T * N_i where N_i has i.i.d. N(0,1) entries, guaranteeing positive definiteness Ellipsoid radius p_i Drawn from U(0, 1)