First-order Conditions Research Articles

Shape and topology optimization for maximum probability domains in quantum chemistry

This article is devoted to the mathematical and numerical treatments of a shape optimization problem emanating from the desire to reconcile quantum theories of chemistry and classical heuristic models: we aim to identify Maximum Probability Domains (MPDs), that is, domains \(\Omega \) of the 3d space where the probability \({\mathbb {P}}_\nu (\Omega )\) to find exactly \(\nu \) among the n constituent electrons of a given molecule is maximum. In the Hartree-Fock framework, the shape functional \({\mathbb {P}}_\nu (\Omega )\) arises as the integral over \(\nu \) copies of \(\Omega \) and \((n-\nu )\) copies of the complement \({\mathbb {R}}^3 \setminus \Omega \) of an analytic function defined over the space \({\mathbb {R}}^{3n}\) of all the spatial configurations of the n electron system. Our first task is to explore the mathematical well-posedness of the shape optimization problem: under mild hypotheses, we prove that global maximizers of the probability functions \({\mathbb {P}}_\nu (\Omega )\) do exist as open subsets of \({\mathbb {R}}^3\); meanwhile, we identify the associated necessary first-order optimality condition. We then turn to the numerical calculation of MPDs, for which we resort to a level set based mesh evolution strategy: the latter allows for the robust tracking of complex evolutions of shapes, while leaving the room for accurate chemical computations, carried out on high-resolution meshes of the optimized shapes. The efficiency of this procedure is enhanced thanks to the addition of a fixed-point strategy inspired from the first-order optimality conditions resulting from our theoretical considerations. Several three-dimensional examples are presented and discussed to appraise the efficiency of our algorithms.

Fracture mechanics: A two-experiment theory

Built into the founding theory of fracture mechanics is a size effect with the size of defects influencing or dictating the structural response of materials. Larger structures can fail at lower loads than might be anticipated from the results obtained from smaller scaled versions made of the same material. The explanation for this phenomenon is that larger defects can exist in the bigger structure and consequently failure loads are correspondingly relatively lower. A slightly different viewpoint is that a single scaled experiment does not adequately capture the response of a defect-ridden structure.This paper introduces a two-experiment theory for fracture mechanics underpinned by experimental tests performed at two distinct scales. It is shown how the size effect associated with defect size is immediately accounted for providing significantly improved representative behaviour than can be otherwise achieved through experiments at a single scale. Two appropriately designed scaled experiments are shown to provide the correct focus in fracture mechanics. It is contended therefore that the current focus on absolute fields (e.g., stress, strain) should be expanded to encompass field differences (or derivatives) with scale in quasistatic fracture mechanics.To demonstrate the importance of field differences classical fracture-mechanics theory is re-examined and stress intensity, J-integrals and cohesive zone models provide the focus. The work builds on the recently discovered finite-similitude theory and on the first-order similitude condition involving two scaled experiments. Standard tests are reanalysed using the finite element method to examine the theory and showcase the potential of the two-experiment approach.

Optimal control of fractional Sturm–Liouville wave equations on a star graph

ABSTRACT In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

Thermo-elastic solid shell formulation with phase field fracture for thin-walled FGMs

Thermo-elastic fracture is a matter of important concern for thin-walled structures made of functionally graded materials (FGMs). Based on this practical relevance, a thermodynamically consistent framework is herein proposed to solve the coupled thermo-mechanical phase-field fracture problem in thin-walled structures made of FGMs. The formulation of the current model is constructed via the evaluation of the phase-field in the Clausius–Duhem inequality leading in to first-order stability conditions in order to ensure thermodynamic consistency. The three-dimensional Kirchhoff–Saint–Venant constitutive model is modified to accommodate the functional grading in the material properties. The computational model is combined with Enhanced Assumed Strain (EAS) and Assumed Natural Strains (ANS) to alleviate locking pathologies concerning the solid shell formulation, leading to a coupled non-linear variational formulation. Several benchmark examples (straight and curved shells) are examined to assess the model capabilities. Moreover, crack deflection, and temperature distributions in the FGM structures are compared with their homogeneous surrogates, to show the importance of the technological solutions with two or even three FGM phases.

Optimal consumption with Hindy–Huang–Kreps preferences under nonlinear expectations

AbstractWe study an intertemporal consumption and portfolio choice problem under Knightian uncertainty in which agent’s preferences exhibit local intertemporal substitution. We also allow for market frictions in the sense that the pricing functional is nonlinear. We prove existence and uniqueness of the optimal consumption plan, and we derive a set of sufficient first-order conditions for optimality. With the help of a backward equation, we are able to determine the structure of optimal consumption plans. We obtain explicit solutions in a stationary setting in which the financial market has different risk premia for short and long positions.

Optimization-based Parametric Model Order Reduction via ${{\mathcal{H}_2} \otimes {\mathcal{L}_2}}$ First-order Necessary Conditions

In this paper, we generalize existing frameworks for $\mathcal{H}_2\otimes\mathcal{L}_2$-optimal model order reduction to a broad class of parametric linear time-invariant systems. To this end, we derive first-order necessary ptimality conditions for a class of structured reduced-order models, and then building on those, propose a stability-preserving optimization-based method for computing locally $\mathcal{H}_2\otimes\mathcal{L}_2$-optimal reduced-order models. We also make a theoretical comparison to existing approaches in the literature, and in numerical experiments, show how our new method, with reasonable computational effort, produces stable optimized reduced-order models with significantly lower approximation errors.

Constrained Trajectory Optimization With Flexible Final Time for Autonomous Vehicles

Differential dynamic programming (DDP) is a well-recognized method for trajectory optimization due to its fast convergence characteristics. However, the original DDP requires a predefined final time and cannot handle nonlinear constraints in optimization. This prohibits the application of DDP for autonomous vehicles due to the heuristic nature of setting a final time beforehand and the existence of inherent physical limits. This article revisits DDP by dynamically optimizing the final time via the first-order optimality condition of the value function and using augmented Lagrangian method to tackle the issue of nonlinear constraints. The resultant algorithm is termed as flexible final time constrained differential dynamic programming (FFT-CDDP). Extensive numerical simulations for a three-dimensional guidance problem are used to demonstrate the working of FFT-CDDP. The results indicate that the proposed FFT-CDDP provides much higher computational efficiency and stronger robustness against the initial solution guess, compared with the off-the-shelf general purpose optimal control software (GPOPS) algorithm.

Online Distributed Optimization With Nonconvex Objective Functions: Sublinearity of First-Order Optimality Condition-Based Regret

In this article, the problem of online distributed optimization with a set constraint is solved by employing a network of agents. Each agent only has access to a local objective function and set constraint, and can only communicate with its neighbors via a digraph, which is not necessarily balanced. Moreover, agents do not have prior knowledge of their future objective functions. Different from existing works on online distributed optimization, we consider the scenario, where objective functions at each time step are nonconvex. To handle this challenge, we propose an online distributed algorithm based on the consensus algorithm and the mirror descent algorithm. Of particular interest is that regrets involving first-order optimality condition are used to measure the performance of the proposed algorithm. Under mild assumptions on the communication graph and objective functions, we prove that regrets grow sublinearly. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.

Sahlqvist correspondence theory for second-order propositional modal logic

AbstractModal logic with propositional quantifiers (i.e. second-order propositional modal logic ($\textsf {SOPML}$)) has been considered since the early time of modal logic. Its expressive power and complexity are high, and its van Benthem–Rosen theorem and Goldblatt–Thomason theorem have been proved by ten Cate (2006, J. Philos. Logic, 35, 209–223). However, the Sahlqvist theory of $\textsf {SOPML}$ has not been considered in the literature. In the present paper, we fill in this gap. We develop the Sahlqvist correspondence theory for $\textsf {SOPML}$, which covers and properly extends existing Sahlqvist formulas in basic modal logic. We define the class of Sahlqvist formulas for $\textsf {SOMPL}$ step by step in a hierarchical way, each formula of which is shown to have a first-order correspondent over Kripke frames effectively computable by an algorithm $\textsf {ALBA}^{\textsf {SOMPL}}$. In addition, we show that certain $\varPi _2$-rules correspond to $\varPi _2$-Sahlqvist formulas in $\textsf {SOMPL}$, which further correspond to first-order conditions, and that even for very simple $\textsf {SOMPL}$ Sahlqvist formulas, they could already be non-canonical.

Optimal H2 Moment Matching-Based Model Reduction for Linear Systems through (Non)convex Optimization

In this paper, we compute a (local) optimal reduced order model that matches a prescribed set of moments of a stable linear time-invariant system of high dimension. We fix the interpolation points and parametrize the models achieving moment-matching in a set of free parameters. Based on the parametrization and using the H2-norm of the approximation error as the objective function, we derive a nonconvex optimization problem, i.e., we search for the optimal free parameters to determine the model yielding the minimal H2-norm of the approximation error. Furthermore, we provide the necessary first-order optimality conditions in terms of the controllability and the observability Gramians of a minimal realization of the error system. We then propose two gradient-type algorithms to compute the (local) optimal models, with mathematical guarantees on the convergence. We also derive convex semidefinite programming relaxations for the nonconvex Problem, under the assumption that the error system admits block-diagonal Gramians, and derive sufficient conditions to guarantee the block diagonalization. The solutions resulting at each step of the proposed algorithms guarantee the achievement of the imposed moment matching conditions. The second gradient-based algorithm exhibits the additional property that, when stopped, yields a stable approximation with a reduced H2-error norm. We illustrate the theory on a CD-player and on a discretized heat equation.

Optimal control problem for a general reaction-diffusion tumor-immune interaction system of mixed immunotherapy and chemotherapy

An optimal control problem for a general reaction-diffusion tumor-immune interaction system under immunotherapy and chemotherapy is discussed to minimize the weighted tumor burden, side effects and treatment costs. The existence, uniqueness and some estimates of strong solution to the state system are obtained by means of the truncation method and semigroup theory. We verify the existence of optimal pair by utilizing the technique of minimizing sequence. The first-order necessary optimality condition and characterization of the optimal control are also derived by proving the differentiability of the control-to-state mapping. In addition, some numerical simulations for this optimal control problem are carried out to present the numerical verification and concrete realization of the theoretical results obtained in this work.

On the generalized trace ratio problem

The generalized trace ratio problem (GTRP) is to maximize the quotient over the Stiefel manifold. We establish the first-order necessary and sufficient optimality conditions for the global maximizer. Based on the classical second-order necessary optimality condition, we prove that GTRP has no local non-global maximizer. Finally we reveal the hidden convexity property of GTRP by applying the convex hull representation of the orthogonal similarity set.

Convergence to a second-order critical point by a primal-dual interior point trust-region method for nonlinear semidefinite programming

In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems, in which the iterates converge to a point that satisfies the first-order and second-order optimality conditions. The method consists of the outer iteration (SDPIP-revised) that finds a Karush-Kuhn-Tucker (KKT) point which satisfies the second-order optimality condition, and the inner iteration (SDPTR-revised) that calculates an approximate barrier KKT point. Algorithm SDPTR-revised uses a commutative class of Newton-like directions within the framework of the trust-region method in the primal-dual space. In addition, we also use a direction of negative curvature when it exists. The proposed algorithm employs a new method that generates negative-curvature directions in the existence of -type penalty term for equality constraints. It is proved that there exists a limit point of the generated sequence which satisfies the second-order optimality condition along with the barrier KKT conditions.

Bilinear Optimal Control of an Advection-Reaction-Diffusion System

We consider the bilinear optimal control of an advection-reaction-diffusion system, where the control arises as the velocity field in the advection term. Such a problem is generally challenging from both theoretical analysis and algorithmic design perspectives, mainly because the state variable depends nonlinearly on the control variable and, an additional divergence-free constraint on the control is coupled together with the state equation. Mathematically, the proof of the existence of optimal solutions is delicate, and, up to now, only some results have been known for a few special cases where additional restrictions are imposed on the space dimension and the regularity of the control. We prove the existence of optimal controls and derive the first-order optimality conditions in general settings without any extra assumptions. Computationally, the well-known conjugate gradient (CG) method can be applied conceptually. However, due to the additional divergence-free constraint on the control variable and the nonlinear relation between the state and control variables, it is challenging to compute the gradient and the optimal stepsize at each CG iteration, and thus nontrivial to implement the CG method. To address these issues, we advocate a fast inner preconditioned CG method to ensure the divergence-free constraint and an efficient inexactness strategy to determine an appropriate stepsize. An easily implementable nested CG method is thus proposed for solving such a complicated problem. For the numerical discretization, we combine finite difference methods for the time discretization and finite element methods for the space discretization. Efficiency of the proposed nested CG method is promisingly validated by the results of some preliminary numerical experiments.

First-order and second-order necessary optimality conditions for discrete-time stochastic systems with delay

Abstract This paper is concerned with the first-order and second-order necessary optimality conditions for discrete-time stochastic systems with delay under weak assumptions. Based on a characteristic method and a discrete-time backward stochastic equation, we establish the linearized discrete-time stochastic maximum principle and Euler-type necessary optimality condition. Moreover, by a new discrete-time backward stochastic matrix equation, we also obtain the necessary optimality condition of quasi-singular control and the pointwise second-order optimality condition.

Application of Pontryagin's maximum principle to quantum metrology in dissipative systems

Optimal control theory, also known as Pontryagin's Maximum Principle, is applied to the quantum parameter estimation in the presence of decoherence. An efficient procedure is devised to compute the gradient of quantum Fisher information with respect to the control parameters and is used to construct the optimal control protocol. The proposed procedure keeps the control problem in the time-invariant form so that both first-order and second-order optimality conditions derived from Pontryagin's Maximum Principle apply; the second-order condition turns out to be crucial when the optimal control contains singular arcs. Concretely we look for the optimal control that maximizes quantum Fisher information for "twist and turn" problem. We find that the optimal control is singular without dissipation but can become unbounded once the quantum decoherence is introduced. An amplitude constraint is needed to guarantee a bounded solution. With quantum decoherence, the maximum quantum Fisher information happens at a finite time due to the decoherence, and the asymptotic value depends on the specific decoherence channel and the control of consideration.

Fast costate estimation for indirect trajectory optimization using Bezier-curve-based shaping approach

In this study, a fast costate estimation method derived from the Bezier-curve-based shaping approach is proposed for indirect low-thrust transfer trajectory optimization. The trajectory is designed rapidly by means of the shape-based method, which can be discretized on the Gauss points and substituted into the discrete first-order optimal condition equation for costate estimation. The estimated costate at the starting point is substituted into the indirect method to solve the optimal control problem. The costate estimation performance for two interplanetary rendezvous missions from Earth to Mars and Earth to Dionysus is compared via random initialization and Newton's iteration method. The initial guesses for the transfer time are given by the Hohmann transfer and the Lambert two-pulse traversal search. The results show that the Lambert traversal search can provide a more accurate launch time estimation in the Bezier-curve-based shaping approach, which can estimate the costate for indirect trajectory optimization in an order of magnitude less time than that required by random initialization.

A penalty decomposition approach for multi-objective cardinality-constrained optimization problems

In this manuscript, we consider multi-objective optimization problems with a cardinality constraint on the vector of decision variables and additional linear constraints. For this class of problems, we analyse necessary and sufficient conditions of Pareto optimality. We afterwards propose a Penalty Decomposition type algorithm, exploiting multi-objective descent methods, to tackle the aforementioned family of problems. We conduct a rigorous convergence analysis for the proposed method, where we prove that the produced sequence of points has limit points, each one being feasible and satisfying first-order optimality conditions. Numerical computational experiments, carried out on instances of relevant real-world problems such as sparse mean/variance portfolio selection and sparse regularized logistic regression, in their multi-objective formulation, show that the proposed procedure is effective at finding solutions forming good Pareto sets approximations.

Bayesian Inference for Risk Minimization via Exponentially Tilted Empirical Likelihood

AbstractThe celebrated Bernstein von-Mises theorem ensures credible regions from a Bayesian posterior to be well-calibrated when the model is correctly-specified, in the frequentist sense that their coverage probabilities tend to the nominal values as data accrue. However, this conventional Bayesian framework is known to lack robustness when the model is misspecified or partly specified, for example, in quantile regression, risk minimization based supervised/unsupervised learning and robust estimation. To alleviate this limitation, we propose a new Bayesian inferential approach that substitutes the (misspecified or partly specified) likelihoods with proper exponentially tilted empirical likelihoods plus a regularization term. Our surrogate empirical likelihood is carefully constructed by using the first-order optimality condition of empirical risk minimization as the moment condition. We show that the Bayesian posterior obtained by combining this surrogate empirical likelihood and a prior is asymptotically close to a normal distribution centering at the empirical risk minimizer with an appropriate sandwich-form covariance matrix. Consequently, the resulting Bayesian credible regions are automatically calibrated to deliver valid uncertainty quantification. Computationally, the proposed method can be easily implemented by Markov Chain Monte Carlo sampling algorithms. Our numerical results show that the proposed method tends to be more accurate than existing state-of-the-art competitors.

Numerical solution to 3D bilinear Fokker-Planck control problem

A characterization and numerical scheme to control problem governed by three-dimensional (3D) time-dependent Fokker-Planck (FP) equation is presented. We formulate a control formulation that controls the drift of the stochastic (FP) process. In this way, the probability density function (PDF) attains a specific configuration. Moreover, a FP control strategy for collective motion is investigated and first-order optimality conditions are presented. On staggered grids, the Chang-Cooper (CC) discretization scheme that ensures the positivity, second-order accuracy, and conservativeness to the FP equation is employed, to the discretized state (resp. adjoint) system. Furthermore, a line search strategy is applied to update the control variable. Results of numerical experiments show the efficiency of the proposed numerical scheme to stochastic (FP) control problems.