First-order Optimality Conditions Research Articles

Adjoint-Based Calibration of Nonlinear Stochastic Differential Equations

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte–Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein–Uhlenbeck and a stochastic Prandtl–Tomlinson bath model are considered.

Moreau-Yosida regularization to optimal control of the monodomain model with pointwise control and state constraints in cardiac electrophysiology

In this work, we study the optimal control problem of a coupled reaction-diffusion system, which is a monodomain model in cardiac electrophysiology with pointwise bilateral control and state constraints. We adopt the Moreau-Yosida regularization as a penalization technique to deal with the state constraints. The regularized problem’s first-order optimality condition is derived. In addition, sufficient second-order optimality condition is derived for the regularized problem using the virtual control concept by proving equivalence between Moreau-Yosida regularization and the virtual control concept. The convergence of optimal controls of the regularized problems to the optimal control of the original problem is proved. Moreover, the semi-smooth Newton method for numerically finding the optimal solution to the regularization problem is presented. Finally, numerical experiments are conducted, and the results allow us to understand the extinction of the wave excitation in cardiac defibrillation in the presence of both control and state constraints.

A Quasi-Newton Subspace Trust Region Algorithm for Nonmonotone Variational Inequalities in Adversarial Learning over Box Constraints

The first-order optimality condition of convexly constrained nonconvex nonconcave min-max optimization problems with box constraints formulates a nonmonotone variational inequality (VI), which is equivalent to a system of nonsmooth equations. In this paper, we propose a quasi-Newton subspace trust region (QNSTR) algorithm for the least squares problems defined by the smoothing approximation of nonsmooth equations. Based on the structure of the nonmonotone VI, we use an adaptive quasi-Newton formula to approximate the Hessian matrix and solve a low-dimensional strongly convex quadratic program with ellipse constraints in a subspace at each step of the QNSTR algorithm efficiently. We prove the global convergence of the QNSTR algorithm to an ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-first-order stationary point of the min-max optimization problem. Moreover, we present numerical results based on the QNSTR algorithm with different subspaces for a mixed generative adversarial networks in eye image segmentation using real data to show the efficiency and effectiveness of the QNSTR algorithm for solving large-scale min-max optimization problems.

On the greenness of separation modes containing compressed fluids

BackgroundIn the past three decades, liquid chromatography (LC) has been recognized as a significant environmental, health, and safety burden due to its heavy reliance on toxic organic solvents. Various chromatographic modes are in vogue today for complex analyses, such as sub/supercritical fluid chromatography (SFC) and enhanced fluidity liquid chromatography (EFLC). These modes are often advertised as "universally green" compared to the traditional allliquid reversed (RPLC) and normal phases (NPLC). Quantitative greenness evaluations must be done to validate or invalidate this assumption and allow separation scientists to make educated choices when deciding on what mode to use. ResultsIn this work, we modify the Analytical Method Greenness Score (AMGS) to include the cycle time of the instrument, and with the help of the first-order optimality condition (by setting the AMGS gradient = 0), we show that SFC and EFLC are not always the greenest option as they are often thought to be. Most of the greenness metrics have ignored the cycle time of instruments, yet this key component changes the entire AMGS response to flow rate. The complex case of separating tobacco alkaloid enantiomers (nicotine, nornicotine, anabasine, and anatabine) was selected as an illustrative example for comparing and contrasting separation modes using the modified greenness metric. These enantiomers have been selected due to their notorious difficulty in separation over the past 30 years. Using this family of molecules, four unique retention patterns were observed covering a wide variety of retention phenomena seen in small molecule enantioseparations. SignificanceThe modified AMGS metric will assist practicing analytical chemists in assessing the environmental impact of their separation methods from a single run in a given chromatographic mode. The proposed methodology identifies the minimum AMGS score corresponding to the greenest separation for routine chemical analysis.

Homotopy trust-region method for phase-field approximations in perimeter-regularized binary optimal control

We consider optimal control problems that have binary-valued control input functions and a perimeter regularization. We develop and analyze a trust-region algorithm that solves a sequence of subproblems in which the regularization term and the binarity constraint are relaxed by a non-convex energy functional. We show how the parameter that controls the distinctiveness of the resulting phase field can be coupled to the trust-region radius updates and be driven to zero over the course of the iterations in order to obtain convergence to points that satisfy a first-order optimality condition of the limit problem under suitable regularity assumptions. Finally, we highlight and discuss the assumptions and restrictions of our approach and provide the first computational results for a motivating application in the field of control of acoustic waves in dissipative media.

Non-coercive Neumann Boundary Control Problems

The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem’s data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included.

Novel efficient iterative schemes for linear finite element approximations of elliptic optimal control problems with integral constraint on the state

Centering on a distributed optimal control problem governed by elliptic equations with a unilateral integral constraint on the state, we recall the first-order optimality conditions to explore variational formulations. The classical linear finite element method is employed to approximate the equivalent formulae. Taking into account the distinctive structural characteristics of discretized schemes, we divide the discretized optimal conditions into two equivalent matrix schemes. Subsequently, we introduce an efficient iterative algorithm designed to solve the approximation schemes. We demonstrate the convergence of our iterative algorithm and investigate its robustness and uniform optimality under a specified constraint, particularly focusing on small regularization parameters. Finally, numerical tests are conducted using various h and α to illustrate the efficiency of our proposed algorithm. These tests also validate a tremendous reduction in the number of required iterations.

Second-order necessary and sufficient optimality conditions for multiobjective interval-valued nonlinear programming

In this article, the second-order optimality conditions for nonlinear programming with multiple interval-valued objective functions (in brief, NPMIOF) are studied. In the first part, the second- and first-order necessary optimality conditions for some types of efficient solutions of NPMIOF are established under of Abadie second- and first-order constraint qualifications. In the next part, we investigate both primal and dual second-order sufficient optimality conditions. Our second-order necessary conditions enhance the previous results. The second-order sufficient optimality conditions are new.

An optimal control strategy to design passive thermal cloaks of arbitrary shape

In this paper, we describe a numerical framework for achieving passive thermal cloaking of arbitrary shapes in both static and transient regimes. The design strategy is cast as the solution of an optimal control problem (OCP) for the heat equation where the coefficients of the thermal diffusivity matrix take the role of control functions, and the distance between the uncloaked and the cloaked field is minimized in a suitable observation domain. The control actions enter bilinearly in the heat equation, thus making the resulting OCP nonlinear, and its analysis non-trivial. We show that optimal diffusivity coefficients exist both for the static and the transient case; we derive a system of first-order necessary optimality conditions; finally, we carry out their numerical approximation using the finite-element method. A series of numerical test cases assess the capability of our strategy to tackle passive thermal cloaking of arbitrarily complex two-dimensional objects.

Loss control regions in optimal control problems

This paper addresses optimal control problems with loss control regions. In that context the state space is partitioned into disjoint subsets, referred to as regions, which are classified into two types: control regions and loss control regions. When the state belongs to a control region, the control is permanent (i.e. the control value is authorized to be modified at any time). On the contrary, when the state belongs to a loss control region, the control must remain constant as long as the state belongs to this region. The objective of this paper is twofold. First, we reformulate the above setting into a hybrid optimal control problem (with spatially heterogeneous dynamics) involving moreover a regionally switching parameter, and we prove a corresponding hybrid maximum principle: hence first-order necessary optimality conditions in a Pontryagin form are obtained. Second, this paper proposes a two-steps numerical scheme to solve optimal control problems with loss control regions. It is based on a direct numerical method (applied to a regularized problem) which initializes an indirect numerical method (applied to the original problem and based on the aforementioned necessary optimality conditions). This numerical approach is applied to several illustrative examples.

Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system’s state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed.

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.

Space-Time Least-Squares Finite Element Methods for Parabolic Distributed Optimal Control Problems

Abstract We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier. Our final formulation fulfills the Babuška–Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem. Numerical experiments confirm our theoretical findings.

An approach to nonlinear optimisation via set stabilisation of dynamical systems with an application to synchronous machines

An approach to constraint nonlinear optimisation is proposed, where the variables of the objective function and of the constraint function belong to the state of a nonlinear dynamical system. A nonlinear closed-loop feedback system is designed, whose solutions converge to a target set consisting of the optimal points of the underlying optimisation problem. The feedback law for the control input of the nonlinear dynamical system is designed by exact linearisation of a dynamical formulation of a first-order optimality condition. For the problem formulation and the stability analysis of the nonlinear closed-loop feedback system, the notion of V-detectability is considered. An application to energy-optimal feed-forward torque control for different types of synchronous machines is presented, which demonstrates the usefulness of the method. In this regard, anisotropic permanent magnet synchronous machines and electrically excited synchronous machines are covered.

Solutions for Poissonian stopping problems of linear diffusions via extremal processes

We develop a general yet simple technique for solving Poissonian timing problems of linear diffusions by relying on the close connection of the extremal processes and the first passage times of the underlying diffusion. We provide a closed-form representation of the expected value gained by employing an ordinary first passage time-based stopping strategy. This approach simplifies the determination of the optimal policy, transforming it into an analysis of ordinary first-order optimality conditions. We relate our findings to various existing approaches for solving stopping problems of linear diffusions and express the optimality conditions in a single boundary setting in a form familiar from optimal stopping of Lévy-processes.

An adaptive phase-field method for structural topology optimization

In this work, we develop an adaptive algorithm for the efficient numerical solution of the minimum compliance problem in topology optimization. The algorithm employs the phase field approximation and continuous density field. The adaptive procedure is driven by two residual type a posteriori error estimators, one for the state variable and the other for the first-order optimality condition of the objective functional. The adaptive algorithm is provably convergent in the sense that the sequence of numerical approximations generated by the adaptive algorithm contains a subsequence convergent to a solution of the continuous first-order optimality system. We provide several numerical simulations to show the distinct features of the algorithm.

On generalized Nash equilibrium problems in infinite-dimensional spaces using Nikaido–Isoda type functionals

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE.

Optimal Sizing of Hybrid Renewable Energy System Using Two-Stage Stochastic Programming

Stochastic programming has become increasingly vital in energy applications, especially in the context of the growing need for renewable energy solutions. This paper presents a significant advancement in this field by introducing an efficient and robust algorithm for optimally sizing hybrid renewable energy systems. Utilizing a two-stage stochastic programming approach, the proposed algorithm addresses the challenges posed by the unpredictability of renewable energy sources. The proposed solution leverages the three-block alternating direction method of multipliers (ADMM), a cutting-edge technique that facilitates parallel computation and enhances computational efficiency. The distinctiveness of this method lies in its ability to solve complex stochastic optimization problems without compromising the mathematical integrity of the model. This is achieved by applying first-order optimality conditions, ensuring both robustness and efficacy. To demonstrate the practical applicability and superiority of the algorithm, a case study was conducted in a rural area of South Africa. The proposed algorithm was applied to design an optimal hybrid renewable energy system, and its performance was compared against traditional methods such as progressive hedging and Monte Carlo techniques. Results affirm the superiority of the approach, saving approximately 8.16% capital cost when compared to progressive hedging. In addition, the proposed algorithm outperforms the Monte Carlo method both in terms of CPU time and the number of cost function evaluations.

Optimal Corrective Maintenance Policies via an Availability-Cost Hybrid Factor for Software Aging Systems

Availability is an important index for the evaluation of the performance of software aging systems. Although the corrective maintenance increases the system availability, the associated cost may be very high; therefore, the balancing of availability and cost during the corrective maintenance phase is a critical issue. This paper investigates optimal corrective maintenance policies via an availability-cost hybrid factor for software aging systems. The system is described by a group of coupled differential equations, where the multiplier effect of the repair rate on a system variable is bilinear term. Our aim is to drive an optimal repair rate that ensures a balance between the maximal system availability and the minimal repair cost. In a finite time interval [0,T], we rigorously discuss the state space of the system and prove the existence of the optimal repair rate, and then derive the first-order necessary optimality conditions by applying a variational inequality with the adjoint variables.

Minimization of integral quadratic estimate of controlled variable in systems with distributed parameters

A constructive method for solving the linear-quadratic problem of optimal control of a parabolic-type system with distributed parameters is proposed under the condition of uniform estimation of target sets. The optimality criterion takes the form of an integral quadratic estimate of the controlled state function in the spatio-temporal domain of its definition. A parameterized representation of control inputs is given with the required accuracy within special intervals of the optimal process, where control inputs cannot be determined using first-order analytical optimality conditions. The suggested approach is based on a previously developed alternance method for constructing parameterized algorithms of programmed control, which heavily relies on fundamental regularities of the subject area. It is demonstrated that the equations of the optimal regulators within the special intervals are reduced to the linear feedback algorithms based on the measured states of the objects. These algorithms are supplemented with switches at boundary points to apply admissible control inputs corresponding to the calculated values of the controlled variable.