Journal of Optimization Theory and Applications最新文献

We consider optimal control problems involving two constraint sets: one comprised of linear ordinary differential equations with the initial and terminal states specified and the other defined by the control variables constrained by simple bounds. When the intersection of these two sets is empty, typically because the bounds on the control variables are too tight, the problem becomes infeasible. In this paper, we prove that, under a controllability assumption, the “best approximation” optimal control minimizing the distance (and thus finding the “gap”) between the two sets is of bang–bang type, with the “gap function” playing the role of a switching function. The critically feasible control solution (the case when one has the smallest control bound for which the problem is feasible) is also shown to be of bang–bang type. We present the full analytical solution for the critically feasible problem involving the (simple but rich enough) double integrator. We illustrate the overall results numerically on various challenging example problems.

We give a general Lagrange multiplier rule for mathematical programming problems in a Hausdorff locally convex space. We consider infinitely many inequality and equality constraints. Our results gives in particular a generalisation of the result of Jahn (Introduction to the theory of nonlinear optimization, Springer, Berlin, 2007), replacing Fréchet-differentiability assumptions on the functions by the Gateaux-differentiability. Moreover, the closed convex cone with a nonempty interior in the constraints is replaced by a strictly general class of closed subsets introduced in the paper and called “admissible sets”. Examples illustrating our results are given.

In this paper, we introduce some constants with the tensors of special structures and present their some useful properties. Furthermore, some perturbation bounds of the tensor complementarity problem are obtained on the base of these constants.

This paper considers the problem of measuring technical efficiency in some class of normed vector spaces. Specifically, the paper focuses on preordered and partially ordered vector spaces by proposing a suitable encompassing netput formulation of the production possibility set. Duality theorems extending some earlier results are established in the context of infinite dimensional spaces. The paper considers directional and normed distance functions and analyzes their relationships. Among other things, overall efficiency can be derived from technical efficiency under a suitable preordered vector space structure. More importantly, it is shown that the existence of core points in partially ordered vector spaces guarantees the comparison of production vectors using the directional distance function. Although the interior of the positive cone may be empty in infinite dimensional vector spaces, it is shown that normed distance functions can also be used to measure efficiency in such spaces by providing them with a suitable preorder structure.

This paper proposes a novel Difference-of-Convex (DC) decomposition for polynomials using a power-sum representation, achieved by solving a sparse linear system. We introduce the Boosted DCA with Exact Line Search ((hbox {BDCA}_text {e})) for addressing linearly constrained polynomial programs within the DC framework. Notably, we demonstrate that the exact line search equates to determining the roots of a univariate polynomial in an interval, with coefficients being computed explicitly based on the power-sum DC decompositions. The subsequential convergence of (hbox {BDCA}_text {e}) to critical points is proven, and its convergence rate under the Kurdyka–Łojasiewicz property is established. To efficiently tackle the convex subproblems, we integrate the Fast Dual Proximal Gradient method by exploiting the separable block structure of the power-sum DC decompositions. We validate our approach through numerical experiments on the Mean–Variance–Skewness–Kurtosis portfolio optimization model and box-constrained polynomial optimization problems. Comparative analysis of (hbox {BDCA}_text {e}) against DCA, BDCA with Armijo line search, UDCA, and UBDCA with projective DC decomposition, alongside standard nonlinear optimization solvers FMINCON and FILTERSD, substantiates the efficiency of our proposed approach.

Nowadays, environmental issues have received increasing attention from experts. The main cause is the increase of carbon emissions in the atmosphere, so it is urgent to reduce carbon emissions. In order to establish the optimal pricing strategy as well as the emission reduction effort strategy for companies who produce and sell low carbon products, this paper proposes an optimal control model for low carbon products. The reduction of the carbon emission for the product is described dynamically by a differential equation, and the analytical expressions of the optimal pricing and the emission abatement strategies are derived using the Pontryagin’s maximum principle. Finally, the numerical experiments are used to explain the results obtained. The results show that companies producing and selling low-carbon products must pay more attention to the amount of carbon emission reduction in their products, and make more efforts to reduce emissions in order to make more profits. Additionally, the parametric analysis shows that expanding market size and reducing inventory depletion can be equally helpful in shortening the sales cycle and boosting profits.

This paper addresses the numerical computation of critical angles between two convex cones in Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary points of a fractional programming problem. To efficiently compute these stationary points, we introduce a partial linearization-like algorithm that offers significant computational advantages. Solving a sequence of strictly convex subproblems with straightforward solutions in several settings gives the proposed algorithm high computational efficiency while delivering reliable results: our theoretical analysis demonstrates that the proposed algorithm asymptotically computes critical angles. Numerical experiments validate the efficiency of our approach, even when dealing with problems of relatively large dimensions: only a few seconds are necessary to compute critical angles between different types of cones in spaces of dimension 1000.

Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared (L^2) norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.

This paper focuses on the stability of solutions for a velocity-tracking problem associated with the two-dimensional Navier–Stokes equations. The considered optimal control problem does not possess any regularizer in the cost, and hence bang-bang solutions can be expected. We investigate perturbations that account for uncertainty in the tracking data and the initial condition of the state, and analyze the convergence rate of solutions when the original problem is regularized by the Tikhonov term. The stability analysis relies on the Hölder subregularity of the optimality mapping, which stems from the necessary conditions of the problem.

In 1963 Boris Polyak suggested a particular step size for gradient descent methods, now known as the Polyak step size, that he later adapted to subgradient methods. The Polyak step size requires knowledge of the optimal value of the minimization problem, which is a strong assumption but one that holds for several important problems. In this paper we extend Polyak’s method to handle constraints and, as a generalization of subgradients, general minorants, which are convex functions that tightly lower bound the objective and constraint functions. We refer to this algorithm as the Polyak Minorant Method (PMM). It is closely related to cutting-plane and bundle methods.