Sufficient Optimality Conditions Research Articles

Optimality conditions for differentiable linearly constrained pseudoconvex programs

AbstractThe aim of this paper is to study optimality conditions for differentiable linearly constrained pseudoconvex programs. The stated results are based on new transversality conditions which can be used instead of complementarity ones. Necessary and sufficient optimality conditions are stated under suitable generalized convexity properties. Moreover, two different pairs of dual problems are proposed and weak and strong duality results proved. Finally, it is shown how transversality conditions can be applied to characterize optimality of convex quadratic problems and to efficiently solve a particular class of Max-Min problems

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

Optimal quadratic full state feedback control of nonlinear affine systems

The solution of the optimal quadratic full-state feedback closed-loop control of a nonlinear affine system problem is presented and solved. This solution uses the State-Dependent Coefficient (SDC) form of a nonlinear system and the Calculus of Variations. Further, it is shown that this solution enables the computation of the value function associated with the respective Hamilton-Jacobi-Bellman equation thus satisfying the necessary and sufficient conditions for optimality. The use of the SDC form representation in the optimal solution for a nonlinear affine system results in an explicit and exact full-state feedback closed-loop control implementation as opposed to the existing open-loop solutions of the optimal control. This solution, like for the linear case, is non-causal and thus cannot be solved in real-time. The closed-loop application involves precomputing a time-varying full-state feedback matrix. The optimal feedback is a linear combination of the current state. The time-varying gain matrix is the backward solution of a non-symmetric matrix Riccati equation. Conditions for the stability of the full-state feedback are presented. This new representation of the optimal solution gives a well-understandable algorithm that enables implementation in the closed loop of the optimal feedback control of a non-linear system which is not possible with the existing optimal open loop representations.

The Optimality Conditions for Generalized Uniform η – V – I Invariant Convexity Programming

Generalized uniformly invariant convex programming is an important optimization problem that has numerous practical applications, such as transportation planning, engineering optimization, etc. The study of its optimal conditions can help solve practical problems and improve efficiency. Optimality conditions serve as the foundation of optimization theory by aiding in understanding the nature and characteristics of problems and providing guidance for the design and analysis of optimization algorithms. Therefore, studying the optimality conditions of generalized uniformly invariant convex programming is highly significant for solving practical problems, promoting the development of optimization theory and algorithm design and analysis, as well as fostering interdisciplinary research. In this research, a novel type of generalized uniform η-V-I invexity function is defined through the use of the η - subdifferential. In the case of the new generalized convex functions, a certain of multiobjective programming with this generalized convexity is discussed, and the corresponding sufficient optimality conditions are obtained.

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

Exact Policy Recovery in Offline RL with Both Heavy-Tailed Rewards and Data Corruption

We study offline reinforcement learning (RL) with heavy-tailed reward distribution and data corruption: (i) Moving beyond subGaussian reward distribution, we allow the rewards to have infinite variances; (ii) We allow corruptions where an attacker can arbitrarily modify a small fraction of the rewards and transitions in the dataset. We first derive a sufficient optimality condition for generalized Pessimistic Value Iteration (PEVI), which allows various estimators with proper confidence bounds and can be applied to multiple learning settings. In order to handle the data corruption and heavy-tailed reward setting, we prove that the trimmed-mean estimation achieves the minimax optimal error rate for robust mean estimation under heavy-tailed distributions. In the PEVI algorithm, we plug in the trimmed mean estimation and the confidence bound to solve the robust offline RL problem. Standard analysis reveals that data corruption induces a bias term in the suboptimality gap, which gives the false impression that any data corruption prevents optimal policy learning. By using the optimality condition for the generalized PEVI, we show that as long as the bias term is less than the ``action gap'', the policy returned by PEVI achieves the optimal value given sufficient data.

Abnormality and Strict-Sense Minimizers That Are Not Extended Minimizers

We consider a constrained optimal control problem and an extension of it, in which the set of strict-sense trajectories is enlarged. Extension is a common procedure in optimal control used to derive necessary and sufficient optimality conditions for the original problem from the extended one, which usually admits a minimizer and has a more regular structure. However, this procedure fails if the two problems have different infima. Therefore, it is relevant to identify such situations. Following on from earlier work by Warga but adopting perturbation techniques developed in nonsmooth analysis, we investigate the relation between the occurrence of an infimum gap and the abnormality of necessary conditions. For the notion of a local minimizer based on control distance and an extension, including the impulsive one, we prove that (i) a local extended minimizer that is not a local minimizer of the original problem, and (ii) a local strict-sense minimizer that is not a local minimizer of the extended problem both satisfy the extended maximum principle in abnormal form. The main novelty is result (ii), as until now, it has only been shown that a strict-sense minimizer that is not an extended minimizer is abnormal for an ‘averaged version’ of the maximum principle.

Extremal Problems for Hyperbolic Systems with Boundary Conditions Involving Integral Time Lags

Extremal problems for integral time lag hyperbolic systems are presented. The optimal boundary control problems for hyperbolic systems in which integral time lags appear in the Neumann boundary conditions are solved. Such systems constitute, in a linear approximation, a universal mathematical model for many processes in which transmission signals at a certain distance with electric, hydraulic and other long lines take place. The time horizon is fixed. Making use of Dubovicki-Milyutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.

Bellman Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost

This work introduces Bellman Neural Networks (BeNNs) and employs them to learn the optimal control actions for the class of optimal control problems (OCPs) with integral quadratic cost. BeNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed and trained to tackle OCPs via applying the Bellman Principle of Optimality (BPO). The BPO provides necessary and sufficient optimality conditions, which result in a nonlinear partial differential equation known as the Hamilton-Jacobi-Bellman (HJB) equation. BeNNs learn the optimal control actions from the unknown solution of the arising HJB equation (i.e., the value function), where the unknown solution is modeled using a Neural Network. Additionally, the paper shows how to estimate the upper bounds on the generalization error of BeNNs while learning the solutions for the OCP class under consideration. The generalization error estimate is provided in terms of the choice and number of the training points as well as the training error. Numerical studies show that BeNNs can be successfully applied to learn the feedback control actions for the class of optimal control problems considered and, after the training is completed, deployed to control the system in a closed-loop fashion. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Impact Statement</i> —The proposed research improves our understanding of how to solve optimal control problems with closed-loop solutions and has potentially a countless number of applications in several different areas. The study is at the intersection between optimal control theory and artificial intelligence connected with mathematical tools for functional interpolation. This advances the ability to implement a higher level of autonomy in decision-making for practical applications with a beneficial impact on our society.

To Charge In-Flight or Not: A Comparison of Two Parallel Hybrid Electric Aircraft Configurations via Optimal Control

This study examines two configurations for parallel hybrid electric aircraft, one with a mechanical connection between the engines and the electric motors and the other without. A previously proposed power allocation algorithm in the context of aircraft energy management is applied to these two configurations for a 19-seat conceptual hybrid electric aircraft. The original optimal control problem is then transformed into a finite-dimensional optimization, and the second-order sufficient conditions for the global optimality of the obtained solution are validated. This is followed by a sensitivity analysis of the fuel consumption to the initial aircraft weight and flight duration. Simulation and theoretical results clarify the limited benefits of charging the battery in-flight for this class of hybrid electric aircraft for the purpose of reducing fuel consumption and CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> emissions.

New second order sufficient optimality conditions for state constrained parabolic control problems

We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local optimality in the sense of L 2 ( Q ) . Our results are valid for spatial domains of dimension less than or equal to three.

Stochastic controls of fractional Brownian motion

Abstract We consider a stochastic control problem for a non-linear forward-backward stochastic differential equation driven by fractional Brownian motion, with Hurst parameter H ∈ ( 0 , 1 ) {H\in(0,1)} , in the case where the set of the control domain is convex. We provide an estimation of the solution and establish the necessary and sufficient optimality conditions in the form of the stochastic maximum principle. We apply the theory to solve a linear quadratic stochastic control problem.

Sufficient optimality, duality and saddle point analysis in terms of convexificators for nonsmooth robust fractional interval-valued optimization problems

In this study, sufficient optimality criteria for a robust fractional interval-valued optimization problem (RP) is highlighted based on the idea of convexificators. The use of generalized invexity tools to support sufficient optimality conditions is demonstrated by an example. Also, saddle point condition of a Lagrange function is examined for RP. Furthermore, Mond–Weir-type robust dual model is constructed and by employing the concept of generalized invexity, usual duality results are discussed.

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

On the Second Order Sufficient Optimality Conditions for a Problem of Mathematical Programming

Duality in the problems of optimal control described by Darboux-type differential inclusions

AbstractThis paper is devoted to the optimization of the Mayer problem with hyperbolic differential inclusions of the Darboux type and duality. We use the discrete approximation method to get sufficient conditions of optimality for the convex problem given by Darboux differential inclusions and the polyhedral problem for a hyperbolic differential inclusion with state constraint. We formulate the adjoint inclusions in the Euler-Lagrange inclusion and Hamiltonian forms. Then, we construct the dual problem to optimal control problem given by Darboux differential inclusions with state constraint and prove so-called duality results. Moreover, we show that each pair of primal and dual problem solutions satisfy duality relations and that the optimal values in the primal convex and dual concave problems are equal. Finally, we establish the dual problem to the polyhedral Darboux problem and provide an example to demonstrate the main constructions of our approach.

The Perturbation Analysis of Nonconvex Low-Rank Matrix Robust Recovery.

In this article, we bring forward a completely perturbed nonconvex Schatten p -minimization to address a model of completely perturbed low-rank matrix recovery (LRMR). This article based on the restricted isometry property (RIP) and the Schatten- p null space property (NSP) generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, and it gives the RIP condition and the Schatten- p NSP assumption that guarantee the recovery of low-rank matrix and the corresponding reconstruction error bounds. In particular, the analysis of the result reveals that in the case that p decreases 0 and for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition (Recht et al., 2010). In addition, we study the connection between RIP and Schatten- p NSP and discern that Schatten- p NSP can be inferred from the RIP. The numerical experiments are conducted to show better performance and provide outperformance of the nonconvex Schatten p -minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.

Optimality conditions associated with new controlled extremization models

&lt;abstract&gt;&lt;p&gt;Applying a parametric approach, in this paper we studied a new class of multidimensional extremization models with data uncertainty. Concretely, first, we derived the robust conditions of necessary optimality. Thereafter, we established robust sufficient optimality conditions by employing the various forms of convexity of the considered functionals. In addition, we formulated an illustrative example to validate the theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Leveraged Matrix Completion With Noise.

Completing low-rank matrices from subsampled measurements has received much attention in the past decade. Existing works indicate that O(nrlog2(n)) datums are required to theoretically secure the completion of an n ×n noisy matrix of rank r with high probability, under some quite restrictive assumptions: 1) the underlying matrix must be incoherent and 2) observations follow the uniform distribution. The restrictiveness is partially due to ignoring the roles of the leverage score and the oracle information of each element. In this article, we employ the leverage scores to characterize the importance of each element and significantly relax assumptions to: 1) not any other structure assumptions are imposed on the underlying low-rank matrix and 2) elements being observed are appropriately dependent on their importance via the leverage score. Under these assumptions, instead of uniform sampling, we devise an ununiform/biased sampling procedure that can reveal the "importance" of each observed element. Our proofs are supported by a novel approach that phrases sufficient optimality conditions based on the Golfing scheme, which would be of independent interest to the wider areas. Theoretical findings show that we can provably recover an unknown n×n matrix of rank r from just about O(nrlog2 (n)) entries, even when the observed entries are corrupted with a small amount of noisy information. The empirical results align precisely with our theories.

Highly efficient scheduling algorithms for identical parallel machines with sufficient conditions for optimality of the solutions

Highly efficient scheduling algorithms for identical parallel machines with sufficient conditions for optimality of the solutions