In this paper, we study the partially positive programming, which is a linear optimization problem over the cone of partially positive matrices with some linear constraints. We reformulate it as a linear programming over the cone of moments, then construct a hierarchy of semidefinite relaxations for solving it, which is built upon the general theory of moment relaxations, flatness and Putinar’s Positivstellensatz. Under mild assumptions, we prove the asymptotic and finite convergence of our semdifinite relaxations. As an application, we consider the mixed binary nonconvex quadratic problem with partially nonnegative variables and reformulate it as different partially positive programmings in two ways. Numerical results show the efficiency of the proposed method.
This paper investigates a new class of stochastic absolute value equations (SAVE), which is equivalent to the stochastic vertical linear complementarity problem. Due to the existence of random variables, the SAVE may not have a common solution for almost every realization. To address this issue, we propose an expected residual minimization (ERM) formulation to seek meaningful solutions to the SAVE. A sample average approximation method based on the Monte Carlo technique is employed to handle the expectation. We establish the coerciveness of the ERM problem and its approximation under the AVE- R_0 pair condition. Moreover, we analyze the convergence of optimal solutions to the approximation problem. Furthermore, by employing the Clarke generalized Jacobian, we study the convergence of stationary points of the ERM problem via constructing a smoothed approximation model. Some numerical examples are reported.
We consider the problem of selecting and scheduling TV advertisements to fill a commercial break as effectively as possible. The gain generated by selecting and airing an ad depends on the ad’s profit and on the position in the break where the ad is scheduled, the latter being represented by an impact function. The resulting problem turns out to be a generalization of the Knapsack Problem, for which we revise and extend results from the literature and show strong NP-hardness in its general form. Then, we focus on the relevant case where the impact function is negative unimodal. For this case, we introduce two new relaxations based on splitting ads into time slices, determine their worst-case optimality gaps, and introduce constructive algorithms for their computation. In addition, a dynamic programming algorithm is introduced for computing an exact solution of the problem. Finally, we present three greedy-type algorithms and analyze their worst-case performance guarantee.
This paper proposes a robust fractional optimal control framework governed by higher-order variational structure and higher-order dynamics. The formulation combines Riemann–Liouville weighted curvilinear action functionals with uncertainty in both the objective and constraints, extending deterministic higher-order variational frameworks to a robust fractional setting. Robust necessary optimality conditions of Euler-Lagrange type are derived, together with a higher-order variational Kuhn-Tucker concept and sufficient conditions for robust weak optimality under convexity assumptions. A numerical example with third-order dynamics illustrates the applicability and accuracy of the proposed approach.
This paper examines structural properties of the solution set of absolute value equations (AVE) of the form Ax + |x| = b . Motivated by recent developments and related open problems in the literature of absolute value equations, and as a continuation of our earlier work [Optimization Letters 20 (2026) 611–623] on the alternative form Ax - |x| = b , we revisit these questions in their original setting. In particular, we analyze solvability of AVE, uniqueness of its solution, computational aspects, as well as its dependence on the properties of the matrix A , and construct counterexamples to show that certain proposed implications of solvability do not hold in general. We then identify sufficient conditions on the matrix A under which these implications become valid, notably establishing that one of them holds when A is symmetric.
