Search Results (81)

With the high penetration of distributed photovoltaics (PV) in distribution areas, transformer capacity limits and source–load fluctuations have become key factors constraining PV accommodation. To accurately assess the PV hosting capacity under energy storage regulation, this paper proposes an assessment method based on operating region analysis. First, a coordinated operation model for the distribution area is established, incorporating the transformer capacity, energy storage constraints, and power balance. On this basis, the calculation boundaries for the PV hosting capacity are discussed in two scenarios: Model 1 ignores power curve uncertainty, characterizing the geometry of the conventional operating region to find the maximum deterministic hosting capacity (S₁) that keeps the region non-empty. Model 2 introduces box-type uncertainty sets for the source and load, proposes the concept of a “Self-Balanced Operating Region”, and constructs a robust feasibility determination model (f₃) based on a Min–Max–Min structure. To solve this multi-layer nested non-convex model, an iterative algorithm based on duality theory and Benders decomposition is employed to determine the robust hosting capacity under uncertainty (S₂) at the critical point where f₃ shifts from zero to non-zero. Case studies show that source–load uncertainty leads to a significant contraction of the operating region, and the robust hosting capacity under uncertainty requirements is strictly less than the deterministic hosting capacity (S₁ > S₂). This method quantifies the reduction effect of uncertainty on the accommodation capability, providing a theoretical basis for planning high-renewable penetration distribution areas and energy storage configuration. Full article

We study Maximum Entropy density estimation on continuous domains under finitely many moment constraints, formulated as the minimization of the Kullback–Leibler divergence with respect to a reference measure. To model uncertainty in empirical moments, constraints are relaxed through convex penalty functions, leading to an infinite-dimensional convex optimization problem over probability densities. The main contribution of this work is a rigorous convex-analytic treatment of such relaxed Maximum Entropy problems in a functional setting, without discretization or smoothness assumptions on the density. Using convex integral functionals and an extension of Fenchel duality, we show that, under mild and explicit qualification conditions, the infinite-dimensional primal problem admits a dual formulation involving only finitely many variables. This reduction can be interpreted as a continuous-domain instance of partially finite convex programming. The resulting dual problem yields explicit primal–dual optimality conditions and characterizes Maximum Entropy solutions in exponential form. The proposed framework unifies exact and relaxed moment constraints, including box and quadratic relaxations, within a single variational formulation, and provides a mathematically sound foundation for relaxed Maximum Entropy methods previously studied mainly in finite or discrete settings. A brief numerical illustration demonstrates the practical tractability of the approach. Full article

The equilibrium shape of a crystal is a fundamental problem in materials science and condensed matter physics. The Wulff construction, a cornerstone of crystal morphology prediction, is traditionally presented and utilized as a powerful geometric algorithm to derive equilibrium shapes from anisotropic surface energy $\gamma \left(\mathbf{n}\right)$. While its application across materials science is vast, the profound mathematical physics underpinning it, specifically its intrinsic identity as a manifestation of the Legendre transform, is often relegated to a passing remark. This work recenters the focus on this fundamental duality. We present a comprehensive, step-by-step derivation of the Wulff shape from the variational principle of surface energy minimization under a constant volume, employing the language of support functions and differential geometry. We then rigorously demonstrate that the equilibrium shape, defined by the support function $h\left(\mathbf{n}\right)$, and the surface energy density $\gamma \left(\mathbf{n}\right)$ are conjugate variables linked by a Legendre transformation; the Wulff shape $\mathcal{W}$ is precisely the zero-sublevel set of the dual function ${\gamma }^{*}\left(\mathbf{x}\right)={sup}_{\mathbf{n}}[\mathbf{x}·\mathbf{n}-\gamma \left(\mathbf{n}\right)]$. This perspective elevates the Wulff construction from a mere graphical tool to a canonical example of convex duality in thermodynamic systems, connecting it to deeper principles in convex analysis and statistical mechanics. To bridge theory and computation, we provide a robust computational algorithm implemented in pseudocode capable of generating Wulff shapes for two-dimensional (2D) crystals with arbitrary N-fold symmetry. Finally, we discuss the relevance and extensions of the classical theory in contemporary research, including non-equilibrium growth, nanoscale effects, and the coupling of crystal shapes with elastic membrane environments. Full article

The reflected–forward–backward splitting (RFBS) method is well-established for solving monotone inclusion problems involving Lipschitz continuous operators in Hilbert spaces, where it converges weakly under mild assumptions. Extending this method to Banach spaces presents significant challenges, primarily due to the nonlinearity of the duality mapping. In this paper, we propose and analyze an RFBS algorithm in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. To the best of our knowledge, this work presents the first strong (R-linear) convergence result for the RFBS method in such Banach spaces, achieved under a newly adapted notion of strong monotonicity. Our results thus establish a foundational theoretical guarantee for RFBS in Banach spaces under strengthened monotonicity conditions, while highlighting the open problem of proving weak convergence for the general monotone case. Full article

The aim of this article is to study duality results for nonsmooth mathematical programs with equilibrium constraints in terms of tangential subdifferentials. We study the Wolfe-type dual problem under the convexity assumptions and a Mond–Weir-type dual problem is also formulated under convexity and generalized convexity assumptions for MPEC by using tangential subdifferentials. We establish weak duality and the two dual programs by assuming tangentially convex functions and also obtain strong duality theorems by assuming generalized standard Abadie constraint qualification. Full article

Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661–674] characterized invertible reverse-ordering transforms in the space of lower, semi-continuous, extended, real-valued convex functions as affine deformations of the ordinary Legendre transform. In this work, we first prove that all those generalized Legendre transforms of functions correspond to the ordinary Legendre transform of dually corresponding affine-deformed functions. In short, generalized convex conjugates are ordinary convex conjugates of dually affine-deformed functions. Second, we explain how these generalized Legendre transforms can be derived from the dual Hessian structures of information geometry. Full article

This paper proves a new lemma that characterizes controllability for linear Volterra control systems and shows that the usual controllability assumption for the variational linearized system near an optimal pair is superfluous. Building on this, it establishes a Pontryagin-type maximum principle for Volterra optimal control with general control and state constraints (fixed terminal constraints and time-dependent state bounds), where the cost combines a terminal term with a state-dependent and integral term. Using the Dubovitskii–Milyutin framework, we construct conic approximations for the cost, dynamics, and constraints and derive necessary optimality conditions under mild regularity: (i) a classical adjoint system when only terminal constraints are present and (ii) a Stieltjes-type adjoint with a non-negative Borel measure when pathwise state constraints are active. Furthermore, under convexity of the cost functional and linear Volterra dynamics, the maximum principle becomes a sufficient criterion for global optimality (recovering the classical sufficiency in the differential case). The differential case recovers the classical PMP, and an SIR example illustrates the results. A key theme is symmetry/duality: the adjoint differentiates in the state while the maximum condition differentiates in the control, reflecting operator transposition and the primal–dual geometry of Dubovitskii–Milyutin cones. Full article

We study a finite-horizon optimal consumption and investment problem in a complete continuous-time market where consumption is restricted within fixed upper and lower bounds. Assuming constant relative risk aversion (CRRA) preferences, we employ the dual-martingale approach to reformulate the problem and derive closed-form integral representations for the dual value function and its derivatives. These results yield explicit feedback formulas for the optimal consumption, portfolio allocation, and wealth processes. We establish the duality theorem linking the primal and dual value functions and verify the regularity and convexity properties of the dual solution. Our results show that the upper and lower consumption bounds transform the linear Merton rule into a piecewise policy: consumption equals L when wealth is low, follows the unconstrained Merton ratio in the interior region, and is capped at H when wealth is high. Full article

This study focuses on the formulation and analysis of problems that are dual to those defined by convex set-valued mappings. Various important classes of optimization problems—such as the classical problems of mathematical and linear programming, as well as extremal problems arising in economic dynamics models—can be reduced to problems of this type. The dual problem proposed in this work is constructed on the basis of the duality theorem connecting the operations of addition and infimal convolution of convex functions, a result that has been previously applied to compact-valued mappings. It appears that, under the so-called nondegeneracy condition, this construction serves as a fundamental approach for deriving duality theorems and establishing both necessary and sufficient optimality conditions. Furthermore, alternative conditions that partially replace the nondegeneracy assumption may also prove valuable for addressing other issues within convex analysis. Full article

The increasing deployment of islanded microgrids in disaster-prone and infrastructure-constrained regions has elevated the importance of resilient energy storage systems capable of supporting autonomous operation. Grid-forming energy storage (GFES) units—designed to provide frequency reference, voltage regulation, and black-start capabilities—are emerging as critical assets for maintaining both energy adequacy and dynamic stability in isolated environments. However, conventional storage planning models fail to capture the interplay between uncertain renewable generation, time-coupled operational constraints, and control-oriented performance metrics such as virtual inertia and voltage ride-through. To address this gap, this paper proposes a novel distributionally robust optimization (DRO) framework that jointly optimizes the siting and sizing of GFES under renewable and load uncertainty. The model is grounded in Wasserstein-metric DRO, allowing worst-case expectation minimization over an ambiguity set constructed from empirical historical data. A multi-period convex formulation is developed that incorporates energy balance, degradation cost, state-of-charge dynamics, black-start reserve margins, and stability-aware constraints. Frequency sensitivity and voltage compliance metrics are explicitly embedded into the optimization, enabling control-aware dispatch and resilience-informed placement of storage assets. A tractable reformulation is achieved using strong duality and solved via a nested column-and-constraint generation algorithm. The framework is validated on a modified IEEE 33-bus distribution network with high PV penetration and heterogeneous demand profiles. Case study results demonstrate that the proposed model reduces worst-case blackout duration by 17.4%, improves voltage recovery speed by 12.9%, and achieves 22.3% higher SoC utilization efficiency compared to deterministic and stochastic baselines. Furthermore, sensitivity analyses reveal that GFES deployment naturally concentrates at nodes with high dynamic control leverage, confirming the effectiveness of the control-informed robust design. This work provides a scalable, data-driven planning tool for resilient microgrid development in the face of deep temporal and structural uncertainty. Full article

The integration of high-penetration renewable energy sources (RESs) into transmission networks introduces profound uncertainty that challenges traditional infrastructure planning approaches. Existing transmission expansion planning (TEP) models either rely on static scenario sets or over-conservative worst-case assumptions, failing to capture the operational stress triggered by rare but structurally impactful renewable behaviors. This paper proposes a novel bi-level optimization framework for transmission planning under adversarial uncertainty, coupling a distributionally robust upper-level investment model with a lower-level operational response embedded with physics and market constraints. The uncertainty space was not exogenously fixed, but instead dynamically generated through a physics-informed spatiotemporal generative adversarial network (PI-ST-GAN), which synthesizes high-risk renewable and load scenarios designed to maximally challenge the system’s resilience. The generator was co-trained using a composite stress index—combining expected energy not served, loss-of-load probability, and marginal congestion cost—ensuring that each scenario reflects both physical plausibility and operational extremity. The resulting bi-level model was reformulated using strong duality, and it was decomposed into a tractable mixed-integer structure with embedded adversarial learning loops. The proposed framework was validated on a modified IEEE 118-bus system with high wind and solar penetration. Results demonstrate that the GAN-enhanced planner consistently outperforms deterministic and stochastic baselines, reducing renewable curtailment by up to 48.7% and load shedding by 62.4% under worst-case realization. Moreover, the stress investment frontier exhibits clear convexity, enabling planners to identify cost-efficient resilience strategies. Spatial congestion maps and scenario risk-density plots further illustrate the ability of adversarial learning to reveal latent structural bottlenecks not captured by conventional methods. This work offers a new methodological paradigm, in which optimization and generative AI co-evolve to produce robust, data-aware, and stress-responsive transmission infrastructure designs. Full article

Over-the-air federated learning (Air-FL) has emerged as a promising paradigm that integrates communication and learning, which offers significant potential to enhance model training efficiency and optimize communication resource utilization. This paper addresses the challenge of interference management in multi-cell Air-FL systems, focusing on parallel multi-task scenarios where each cell independently executes distinct training tasks. We begin by analyzing the impact of aggregation errors on local model performance within each cell, aiming to minimize the cumulative optimality gap across all cells. To this end, we formulate an optimization framework that jointly optimizes device transmit power and denoising factors. Leveraging the Pareto boundary theory, we design a centralized optimization scheme that characterizes the trade-offs in system performance. Building upon this, we propose a distributed power control optimization scheme based on interference temperature (IT). This approach decomposes the globally coupled problem into locally solvable subproblems, thereby enabling each cell to adjust its transmit power independently using only local channel state information (CSI). To tackle the non-convexity inherent in these subproblems, we first transform them into convex problems and then develop an analytical solution framework grounded in Lagrangian duality theory. Coupled with a dynamic IT update mechanism, our method iteratively approximates the Pareto optimal boundary. The simulation results demonstrate that the proposed scheme outperforms baseline methods in terms of training convergence speed, cross-cell performance balance, and test accuracy. Moreover, it achieves stable convergence within a limited number of iterations, which validates its practicality and effectiveness in multi-task edge intelligence systems. Full article

This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided. Full article

In urban environments, the highly complex communication environment often leads to blockages in the link between ground users (GUs) and unmanned aerial vehicles (UAVs), resulting in poor communication quality. Although traditional reconfigurable intelligent surfaces (RISs) can improve wireless channel quality, they can only provide reflection services and have limited coverage. For this reason, we study a novel simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR–RIS)–assisted UAV–mobile edge computing (UAV–MEC) network, which can serve multiple users residing in the transmission area and reflection area, and switch between reflection and transmission modes according to the relative positions of the UAV, GUs, and STAR–RIS, providing users with more flexible and efficient services. The system comprehensively considers user transmit power, time slot allocation, UAV flight trajectory, STAR–RIS mode selection, and phase angle matrix, achieving long–term energy consumpution minimization while ensuring stable task backlog queue. Since the proposed problem is a long–term stochastic optimization problem, we use the Lyapunov method to transform it into three deterministic online optimization subproblems and iteratively solve them alternately. Specifically, we firstly use the Lambert function to solve for the closed-form solution of the transmit power; then, use Lagrange duality and the Karush–Kuhn–Tucker conditions to solve time slot allocation; finally, successive convex approximation is used to obtain trajectory planning for UAVs with lower complexity, and triangular inequalities are used to solve the STAR–RIS phase shift. The simulation results show that the proposed scheme has better performance than other benchmark schemes in maintaining queue stability and reducing energy consumption. Full article

In this paper, a class of controlled variational control models is studied by considering the notion of $(q,w)-\pi$-invexity. Our aim is to investigate a solution set in the considered interval-valued controlled models. To achieve this, we establish some characterization results of solutions in the controlled interval-valued variational models. More precisely, necessary and sufficient conditions of optimality are highlighted as part of a feasible solution. To prove that the optimality conditions are sufficient, we impose generalized invariant convexity hypotheses for the involved multiple integral functionals. Finally, a duality result is provided in order to better describe the problem under study. The methodology used in this paper is a combination of techniques from the Lagrange–Hamilton theory, calculus of variations, and control theory. This study could be immediately improved by including an analysis of this class of optimization problems by using curvilinear integrals instead of multiple integrals. The independence of path imposed to these functionals and their physical significance would increase the applicability and importance of the paper. Full article