Infinite-dimensional Optimization Problem Research Articles

A slope-based automatic identification and optimization of key time nodes method for adaptive control vector parameterization

Control vector parameterization is a mainstream numerical approach for solving dynamic programming problems. By discretizing the entire time domain into a grid of time nodes, an infinite-dimensional optimal control problem can be transformed into a finite-dimensional static optimization problem. Despite the attractive advantages of high solution accuracy and ease of implementation of control vector parameterization, the rationality of time grid partitioning significantly influences the efficiency of the solution and the approximating accuracy of the optimal control trajectory. In the traditional control vector parameterization method, the time grid is typically set beforehand and remains static in the optimization process, which has a direct impact on how closely the solution result approximates the optimal control trajectory. To address the conflict of accurate approximations and computational time, we propose a slope-based automatic identification and optimization of key time nodes method for adaptive control vector parameterization, which not only optimizes merging and inserting time nodes but also incorporates automatic identification of key time nodes. Through a simulation example, we verify that this improved method can reduce computation time, enhance approximation accuracy, and achieve higher optimization efficiency.

Robust support function machines for set-valued data classification

Support function machines (SFMs) have been proposed to handle set-valued data, but they are sensitive to outliers and unstable for re-sampling due to the use of the hinge loss function. To address these problems, we propose a robust SFM model with proximity functions. We first define a family of proximity functions that are used to convert set-valued data into continuous functions in a Banach space, and then we use the margin maximization in a Banach space to construct the pinball SFMs (PinSFMs). We study some properties of the proposed model, and it is interesting to observe that the optimal measure of the proposed model has a specific representation under the total variation norm. Using the representation of the optimal measure, we convert an infinite-dimensional optimization problem into a finite-dimensional optimization problem. Unlike SFMs, we employ a sampling strategy to tackle the finite-dimensional optimization problem. We theoretically show that the sparse solution determines the sparsity of the sampling points though the sampling strategy causes uncertainty for the sampling points. In addition, we achieve kernel versions of proximity functions, and the attractive property of this kernelization is that the proposed model is convex even if indefinite kernels are employed. Experiments on a series of data sets are performed to demonstrate that the proposed model is superior to some existing models in the presence of outliers.

Information Design and Sharing in Supply Chains

We study the interplay between inventory replenishment policies and information sharing in the context of a two-tier supply chain with a single supplier and a single retailer serving an independent and identically distributed Gaussian market demand. We investigate how the retailer’s inventory policy impacts the supply chain’s cumulative expected long-term average inventory costs [Formula: see text] in two extreme information-sharing cases: (a) full information sharing and (b) no information sharing. To find the retailer’s inventory policy that minimizes [Formula: see text], we formulate an infinite-dimensional optimization problem whose decision variables are the MA([Formula: see text]) coefficients that characterize a stationary ordering policy. Under full information sharing, the optimization problem admits a simple solution and the optimal policy is given by an MA(1) process. On the other hand, to solve the optimization problem under no information sharing, we reformulate the optimization from its time domain formulation to an equivalent z-transform formulation in which the decision variables correspond to elements of the Hardy space H2. This alternative representation allows us to use a number of results from H2 theory to compute the optimal value of [Formula: see text] and characterize a sequence of ϵ-optimal inventory policies under some mild technical conditions. By comparing the optimal solution under full information sharing and no information sharing, we derive a number of important practical takeaways. For instance, we show that there is value in information sharing if and only if the retailer’s optimal policy under full information sharing is not invertible with respect to the sequence of demand shocks. Furthermore, we derive a fundamental mathematical identity that reveals the value of information sharing by exploiting the canonical Smirnov–Beurling inner–outer factorization of the retailer’s orders when viewed as an element of H2. We also show that the value of information sharing can grow unboundedly when the cumulative supply chain costs are dominated by the supplier’s inventory costs. Funding: R. Caldentey acknowledges the University of Chicago Booth School of Business for financial support. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.008 .

Trajectory generation for the unicycle model using semidefinite relaxations

We develop a convex relaxation for the minimum energy control problem of the well-known unicycle model in the form of a semidefinite program. Through polynomialization techniques, the infinite-dimensional optimal control problem is first reformulated as a non-convex, infinite-dimensional quadratic program which can be viewed as a trajectory generation problem. This problem is then discretized to arrive at a finite-dimensional, albeit, non-convex quadratically constrained quadratic program. By applying the moment relaxation method to this quadratic program, we obtain a hierarchy of semidefinite relaxations. We construct an approximate solution for the infinite-dimensional trajectory generation problem by solving the first- or second-order moment relaxation. A comprehensive simulation study provided in this paper suggests that the second-order moment relaxation is lossless.

Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

Single-Projection Procedure for Infinite Dimensional Convex Optimization Problems

Worst-case risk with unspecified risk preferences

In this paper, we study the worst-case distortion risk measure for a given risk when information about distortion functions is partially available. We obtain the explicit forms of the worst-case distortion functions for several different sets of plausible distortion functions. When there is no concavity constraint on distortion functions, the worst-case distortion function is independent of the risk to be measured and the corresponding worst-case distortion risk measure is the weighted average of the VaR's of the risk for all decision makers. When the concavity constraint is imposed on distortion functions and the set of concave distortion functions is defined by the riskiness of one single risk, the explicit form of the worst-case distortion function is obtained, which depends the risk to be measured. When the set of concave distortion functions is defined by the riskiness of multiple risks, we reduce the infinite-dimensional optimization problem to a finite-dimensional optimization problem which can be solved numerically. Finally, we apply the worst-case risk measure to optimal decision making in reinsurance.

A Formal Characterization of Activation Functions in Deep Neural Networks.

In this article, a mathematical formulation for describing and designing activation functions in deep neural networks is provided. The methodology is based on a precise characterization of the desired activation functions that satisfy particular criteria, including circumventing vanishing or exploding gradients during training. The problem of finding desired activation functions is formulated as an infinite-dimensional optimization problem, which is later relaxed to solving a partial differential equation. Furthermore, bounds that guarantee the optimality of the designed activation function are provided. Relevant examples with some state-of-the-art activation functions are provided to illustrate the methodology.

Stability of Solutions to Extremal Problems with Constraints Based on λ-Truncations

In this paper, we consider finite- and infinite-dimensional optimization problems with constraints of general type. We obtain sufficient conditions for stability of a strict solution and conditions for stability of a set of solutions with more than one point in it according to small perturbations of the problem parameters. For finite-dimensional extremal problems with equality-type constraints, we obtain stability conditions based on the construction of λ-truncations of mappings.

On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation (PDE) which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc(ℝ). This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. The convergence of the resulting algorithm is proven in the function space setting. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero. The paper concludes with numerical experiments which confirm the theoretical findings.

The Competition Complexity of Dynamic Pricing

We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward [Formula: see text] achievable by the optimal online policy on m independent and identically distributed (i.i.d.) random variables distributed according to F to the expected maximum [Formula: see text] of n i.i.d. draws from F. We ask how big m has to be to ensure that [Formula: see text] for all F. We resolve this question and characterize the competition complexity as a function of ε. When [Formula: see text], the competition complexity is unbounded. That is, for any n and m there is a distribution F such that [Formula: see text]. In contrast, for any [Formula: see text], it is sufficient and necessary to have [Formula: see text], where [Formula: see text]. Therefore, the competition complexity not only drops from unbounded to linear, it is actually linear with a very small constant. The technical core of our analysis is a lossless reduction to an infinite dimensional and nonlinear optimization problem that we solve optimally. A corollary of this reduction is a novel proof of the factor [Formula: see text] i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds. Funding: This work was supported by ANID (Anillo ICMD) [Grant ACT210005] and the Center for Mathematical Modeling [Grant FB210005].

Group sparse optimization for inpainting of random fields on the sphere

Abstract We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed optimization model, the variable is an infinite-dimensional complex vector and the objective function is a real-valued function defined by a hybrid of the $\ell _2$ norm and non-Lipschitz $\ell _p (0<p<1)$ norm that preserves rotational invariance property and group structure of the random complex coefficients. We show that the infinite-dimensional optimization problem is equivalent to a convexly-constrained finite-dimensional optimization problem. Moreover, we propose a smoothing penalty algorithm to solve the finite-dimensional problem via unconstrained optimization problems. We provide an approximation error bound of the inpainted random field defined by a scaled Karush–Kuhn–Tucker (KKT) point of the constrained optimization problem in the square-integrable space on the sphere with probability measure. Finally, we conduct numerical experiments on band-limited random fields on the sphere and images from Cosmic Microwave Background (CMB) data to show the promising performance of the smoothing penalty algorithm for inpainting of random fields on the sphere.

Convex Approach to Data-Driven Off-Road Navigation via Linear Transfer Operators

We consider the problem of optimal control design for navigation on off-road terrain. We use a traversability measure to characterize the difficulty of navigation on off-road terrain. The traversability measure captures terrain properties essential for navigation, such as elevation maps, roughness, slope, and texture. The terrain with the presence or absence of obstacles becomes a particular case of the proposed traversability measure. We provide a convex formulation to the off-road navigation problem by lifting the problem to the density space using the linear Perron-Frobenius (P-F) operator. The convex formulation leads to an infinite-dimensional optimal navigation problem for control synthesis. We construct the finite-dimensional approximation of the optimization problem using data. We use a computational framework based on the data-driven approximation of the Koopman operator. This makes the proposed approach data-driven and applicable to cases where an explicit system model is unavailable. Finally, we apply the proposed navigation framework with single integrator dynamics and Dubin's car model.

Numerical Methods for Fractional Optimal Control and Estimation

Fractional calculus has become a valuable mathematical tool for modeling various physical phenomena exhibiting anomalous dynamics such as memory and hereditary properties. However, the fractional operators lead to difficulties in analysis, optimization, and estimation that limit the application of fractional models. This paper develops numerical methods to solve fractional optimal control and estimation problems with Caputo derivatives of arbitrary order.
 First, fractional Pontryagin's maximum principle is used to formulate first-order necessary conditions for fractional optimal control problems. A fractional collocation method using polynomial basis functions is then proposed to discretize the resulting boundary value problems. This allows transforming an infinite-dimensional optimal control problem into a finite nonlinear programming problem.
 Second, for fractional estimation, a novel ensemble Kalman filter is proposed based on a Monte Carlo approach to propagate the fractional state dynamics. This provides a recursive fractional state estimator analogous to the classical Kalman filter.
 The capabilities of the proposed collocation and ensemble Kalman filter methods are demonstrated through applications including fractional epidemic control, thermomechanical oscillator control, and state estimation of viscoelastic mechanical systems. The results illustrate improved accuracy over prior discretization schemes along with the ability to handle complex system dynamics.
 This work provides a comprehensive framework for numerical solution of fractional optimal control and estimation problems. The methods enable applying fractional calculus to address challenges in robotics, biomedicine, mechanics, and other fields where systems exhibit non-classical dynamics.

A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. Measurability and convergence rates of a stationarity measure are handled, filling a gap for applications to nonconvex infinite dimensional stochastic optimization problems. The method is demonstrated on an optimal control problem constrained by a class of elliptic semilinear partial differential equations (PDEs) under uncertainty.

Machine-Synthesized Control of Nonlinear Dynamic Object Based on Optimal Positioning of Equilibrium Points

When solving an optimal control problem with both direct and indirect approaches, the main technique is to transfer the optimal control problem from the class of infinite-dimensional optimization to a finite-dimensional one. However, with all these approaches, the result is an open-loop program control that is sensitive to uncertainties, and for the implementation of which in a real object it is necessary to build a stabilization system. The introduction of the stabilization system changes the dynamics of the object, which means that the optimal control and the optimal trajectory should be calculated for the object already taking into account the stabilization system. As a result, it turns out that the initial optimal control problem is complex, and often the possibility of solving it is extremely dependent on the type of object and functionality, and if the object becomes more complex due to the introduction of a stabilization system, the complexity of the problem increases significantly and the application of classical approaches to solving the optimal control problem turns out to be time-consuming or impossible. In this paper, a synthesized optimal control method is proposed that implements the designated logic for developing optimal control systems, overcoming the computational complexity of the problem posed through the use of modern machine learning methods based on symbolic regression and evolutionary optimization algorithms. According to the approach, the object stabilization system is first built relative to some point, and then the position of this equilibrium point becomes a control parameter. Thus, it is possible to translate the infinite-dimensional optimization problem into a finite-dimensional optimization problem, namely, the optimal location of equilibrium points. The effectiveness of the approach is demonstrated by solving the problem of optimal control of a mobile robot.

A Necessary Optimality Condition for Optimal Control of Caputo Fractional Evolution Equations

In this paper, we prove the Pontryagin maximum principle, which constitutes the necessary optimality condition, for the infinite-dimensional optimal control problem of X-valued left Caputo fractional evolution equations, where X is a Banach space. An important step in the proof to obtain the desired Hamiltonian maximization condition is to establish new variational and duality analysis. While the former is characterized by a linear X-valued left Caputo fractional evolution equation via spike variation, the latter requires the adjoint equation characterized by a linear X*-valued right Riemann-Liouville (RL) fractional evolution equation, where X* is a dual space of X. We show the variational and duality analysis with the help of the infinite-dimensional fractional version of the technical lemma and the explicit representation of solutions to linear (Caputo and RL) fractional evolution equations using left and right RL state-transition evolution operators.

Distributionally robust reinsurance with expectile

AbstractWe study a distributionally robust reinsurance problem with the risk measure being an expectile and under expected value premium principle. The mean and variance of the ground-up loss are known, but the loss distribution is otherwise unspecified. A minimax problem is formulated with its inner problem being a maximization problem over all distributions with known mean and variance. We show that the inner problem is equivalent to maximizing the problem over three-point distributions, reducing the infinite-dimensional optimization problem to a finite-dimensional optimization problem. The finite-dimensional optimization problem can be solved numerically. Numerical examples are given to study the impacts of the parameters involved.

Control in Probability for SDE Models of Growth Population.

In this paper, we consider a (control) optimization problem, which involves a stochastic dynamic. The model proposes selecting the best control function that keeps bounded a stochastic process over an interval of time with a high probability level. Here, the stochastic process is governed by a stochastic differential equation affected by a stochastic process. This setting becomes a chance-constrained control optimization problem, where the constraint is given by the probability level of infinitely many random inequalities. Since such a model is challenging, we discretize the dynamic and restrict the space of control functions to piecewise mappings. On the one hand, it transforms the infinite-dimensional optimization problem into a finite-dimensional one. On the other hand, it allows us to provide the well-posedness of the problem and approximation. Finally, the results are illustrated with numerical results, where classical model for the growth of a population are considered.

Sample average approximations of strongly convex stochastic programs in Hilbert spaces

We analyze the tail behavior of solutions to sample average approximations (SAAs) of stochastic programs posed in Hilbert spaces. We require that the integrand be strongly convex with the same convexity parameter for each realization. Combined with a standard condition from the literature on stochastic programming, we establish non-asymptotic exponential tail bounds for the distance between the SAA solutions and the stochastic program’s solution, without assuming compactness of the feasible set. Our assumptions are verified on a class of infinite-dimensional optimization problems governed by affine-linear partial differential equations with random inputs. We present numerical results illustrating our theoretical findings.

Random field optimization

We present a new modeling paradigm for optimization that we call random field optimization. Random fields are a powerful modeling abstraction that aims to capture the behavior of random variables that live on infinite-dimensional spaces (e.g., space and time) such as stochastic processes (e.g., time series, Gaussian processes, and Markov processes), random matrices, and random spatial fields. This paradigm involves sophisticated mathematical objects (e.g., stochastic differential equations and space-time kernel functions) and has been widely used in neuroscience, geoscience, physics, civil engineering, and computer graphics. Despite of this, however, random fields have seen limited use in optimization; specifically, existing optimization paradigms that involve uncertainty (e.g., stochastic programming and robust optimization) mostly focus on the use of finite random variables. This trend is rapidly changing with the advent of statistical optimization (e.g., Bayesian optimization) and multi-scale optimization (e.g., integration of molecular sciences and process engineering). Our work extends a recently-proposed abstraction for infinite-dimensional optimization problems by capturing more general uncertainty representations. Moreover, we discuss solution paradigms for this new class of problems based on finite transformations and sampling, and identify open questions and challenges.