Journal of Optimization Theory and Applications最新文献

Given a nonlinear control system, a target set, a nonnegative integral cost, and a continuous function W, we say that the system is globally asymptotically controllable to the target with W-regulated cost, whenever, starting from any point z, among the strategies that achieve classical asymptotic controllability we can select one that also keeps the cost less than W(z). In this paper, assuming mild regularity hypotheses on the data, we prove that a necessary and sufficient condition for global asymptotic controllability with regulated cost is the existence of a special, continuous Control Lyapunov Function, called a Minimum Restraint Function. The main novelty is the necessity implication, obtained here for the first time. Nevertheless, the sufficiency condition extends previous results based on semiconcavity of the Minimum Restraint Function, while we require mere continuity.

We study a class of mean-field games with incomplete information in this paper. For each agent, the state is given by a linear forward stochastic differential equation with common noise. Moreover, both the state and control variables can enter the diffusion coefficients of the state equation. We deduce the open-loop adapted decentralized strategies and feedback decentralized strategies by a mean-field forward–backward stochastic differential equation and Riccati equations, respectively. The well-posedness of the corresponding consistency condition system is obtained and the limiting state-average turns out to be the solution of a mean-field stochastic differential equation driven by common noise. We also verify the (varepsilon )-Nash equilibrium property of the decentralized strategies. Finally, a network security problem is studied to illustrate our results as an application.

We consider a class of time-inhomogeneous optimal stopping problems and we provide sufficient conditions on the data of the problem that guarantee monotonicity of the optimal stopping boundary. In our setting, time-inhomogeneity stems not only from the reward function but, in particular, from the time dependence of the drift coefficient of the one-dimensional stochastic differential equation (SDE) which drives the stopping problem. In order to obtain our results, we mostly employ probabilistic arguments: we use a comparison principle between solutions of the SDE computed at different starting times, and martingale methods of optimal stopping theory. We also show a variant of the main theorem, which weakens one of the assumptions and additionally relies on the renowned connection between optimal stopping and free-boundary problems.

In this paper, we establish new necessary and sufficient conditions guaranteeing the uniform LP duality for linear problems of Copositive Programming and formulate these conditions in different equivalent forms. The main results are obtained using the approach developed in previous papers of the authors and based on a concept of immobile indices that permits alternative representations of the set of feasible solutions.

Local solutions for variational and quasi-variational inequalities are usually the best type of solutions that could practically be obtained when in lack of convexity or else when available numerical techniques are too limited for global solutions. Nevertheless, the analysis of such problems found in the literature seems to be very restricted to the global treatment. Motivated by this fact, in this work, we propose local solution concepts, study their interrelations and relations with global concepts and prove existence results as well as stability of local solution map of parametric variational inequalities. The key ingredient of our results is the new concept of local reproducibility of a set-valued map, which we introduce to explore such local solutions to quasi-variational inequality problems. As a by-product, we obtain local solutions to quasi-optimization problems, bilevel quasi-optimization problems and Single-Leader-Multi-Follower games.

Below we derive necessary conditions of optimality for problems with mixed constraints (see Dmitruk in Control Cybern 38(4A):923–957, 2009) using the method of penalty functions similar to the one we previously used to solve optimization problems for control sweeping processes (see, e.g., De Pinho et al. in Optimization 71(11):3363–3381, 2022) and, more recently, to solve optimal control problems with pure state constraints (see De Pinho et al. in Syst Control Lett 188:105816, 2024). We intentionally consider a smooth case and the simplest boundary conditions; we consider global minimum and assume that the set of trajectories of the control system is compact. Based on our penalty functions approach we develop a numerical method admitting estimates for its parameters needed to achieve a given precision.

We study the approximate controllability of the discrete fractional systems of order (1<alpha <2)

subject to the initial states (u^0=x_0,u^1=x_1,) where A is a closed linear operator defined in a Hilbert space X, B is a bounded linear operator from a Hilbert space U into (X, f:{mathbb {N}}_0times Xrightarrow X) is a given sequence and (,_Cnabla ^{alpha } u^n) is an approximation of the Caputo fractional derivative (partial ^alpha _t) of u at (t_n:=tau n,) where (tau >0) is a given step size. To do this, we first study resolvent sequences ({S_{alpha ,beta }^n}_{nin {mathbb {N}}_0}) generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to ((*)) and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system ((*)) in terms of the resolvent sequence ({S_{alpha ,beta }^n}_{nin {mathbb {N}}_0}) for some (alpha ,beta >0.) Finally, we provide an example to illustrate our results.

Hybrid renewable energy systems (HRESs) that integrate conventional and renewable energy generation and energy storage technologies represent a viable option to serve the energy demand of remote and isolated communities. A common way to capture the stochastic nature of demand and renewable energy supply in such systems is by using a small number of independent discrete scenarios. However, some information is inevitably lost when extracting these scenarios from historical data, thus introducing errors and biases to the design process. This paper proposes two frameworks, namely robust-stochastic optimization and distributionally robust optimization, that aim to hedge against the resulting uncertainty of scenario characterization and probability, respectively, in scenario-based HRES design approaches. Mathematical formulations are provided for the nominal, stochastic, robust-stochastic, distributional robust, and combined problems, and directly-solvable tractable reformulations are derived for the stochastic and the distributional robust cases. Furthermore, an exact column-and-constraint-generation algorithm is developed for the robust-stochastic and combined cases. Numerical results obtained from a realistic case study of a stand-alone solar-wind-battery-diesel HRES serving a small community in Northern Ontario, Canada reveal the performance advantage, in terms of both cost and utilization of renewable sources, of the proposed frameworks compared to classical deterministic and stochastic models, and their ability to mitigate the issue of information loss due to scenario reduction.

Using the theory of Hilbert direct integrals, we introduce and study a monotonicity-preserving operation, termed the integral resolvent mixture. It combines arbitrary families of monotone operators acting on different spaces and linear operators. As a special case, we investigate the resolvent expectation, an operation which combines monotone operators in such a way that the resulting resolvent is the Lebesgue expectation of the individual resolvents. Along the same lines, we introduce an operation that mixes arbitrary families of convex functions defined on different spaces and linear operators to create a composite convex function. Such constructs have so far been limited to finite families of operators and functions. The subdifferential of the integral proximal mixture is shown to be the integral resolvent mixture of the individual subdifferentials. Applications to the relaxation of systems of composite monotone inclusions are presented.

The polynomial complementarity problem (PCP) is an important extension of the tensor complementarity problem (TCP). The main purpose of the present paper is to extend the results on the bounds of solutions of TCP due to Xu–Gu–Huang from TCP to PCP. To that end, the concepts of (generalized) row strictly diagonally dominant tensor to tensor tuple are extended and the properties about them are discussed. By using the introduced structured tensor tuples, the upper and lower bounds on the norm of solutions to PCP are derived. Comparisons between the results presented in the present paper and the existing bounds are made.