Finite-dimensional Optimization 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.

Robustifying Conditional Portfolio Decisions via Optimal Transport

We propose a data-driven portfolio selection model that integrates side information, conditional estimation, and robustness using the framework of distributionally robust optimization. Conditioning on the observed side information, the portfolio manager solves an allocation problem that minimizes the worst-case conditional risk-return tradeoff, subject to all possible perturbations of the covariate-return probability distribution in an optimal transport ambiguity set. Despite the nonlinearity of the objective function in the probability measure, we show that the distributionally robust portfolio allocation with a side information problem can be reformulated as a finite-dimensional optimization problem. If portfolio decisions are made based on either the mean-variance or the mean-conditional value-at-risk criterion, the reformulation can be further simplified to second-order or semidefinite cone programs. Empirical studies in the U.S. equity market demonstrate the advantage of our integrative framework against other benchmarks. Funding: The material in this paper is based on work supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397]. Additional support is gratefully acknowledged from the National Science Foundation [Grants 1915967, 1820942, and 1838676], the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2016-05208], and the China Merchant Bank. V. A. Nguyen gratefully acknowledges the generous support from the Chinese University of Hong Kong [Improvement on Competitiveness in Hiring New Faculties Funding Scheme] and the Chinese University of Hong Kong [Direct Grant 4055191]. S. Wang is partially supported by the National Natural Science Foundation of China [Grant 72371022]. Finally, this research was enabled in part by support provided by Compute Canada. Supplemental Material: The computer code and data that support the findings of this study and the online appendix are available within this article’s supplemental material at https://doi.org/10.1287/opre.2021.0243 .

Deterministic scenarios guided [formula omitted]-Adaptability in multistage robust optimization for energy management and cleaning scheduling of heat transfer process

The heat transfer process is of paramount importance for energy management and heat recovery, but the inevitable uncertain fouling poses significant challenges to sustainable energy-saving efforts. This study aims to solve the energy management and cleaning scheduling problems under fouling uncertainty. To achieve this, a novel multistage Robust Optimization (RO) method based on the Deterministic Scenarios Guided K-Adaptability (DSGKA) strategy is proposed. The process scheduling problem is initially tackled through long-period Mixed Integer Optimal Control Problems (MIOCPs) with deterministic scenarios. Subsequently, considering the slow transition characteristics of energy efficiency degradation caused by fouling, the candidate paths generation algorithm guided by deterministic MIOCPs is developed. Additionally, the K-Adaptability method is employed to segment decision-dependent uncertainty sets into finite partitions, facilitating the resolution of the multistage RO problem through worst-case analysis. In theory, the rationality of the proposed guidance algorithm is established. Simulation results on a heat exchanger network are also provided to demonstrate the effectiveness of the DSGKA scheme. By transforming the original multistage RO problem into a finite-dimensional optimization problem solvable with intelligent heuristic algorithms, the proposed method adeptly manages nonlinearity in constraint equations, thereby improving its applicability for equipment maintenance scheduling across various slow transition industrial processes.

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.

Stabilized SQP Methods in Hilbert Spaces

Based on techniques by (S.J. Wright 1998) for finite-dimensional optimization, we investigate a stabilized sequential quadratic programming method for nonlinear optimization problems in infinite-dimensional Hilbert spaces. The method is shown to achieve fast local convergence even in the absence of a constraint qualification, generalizing the results obtained by (S.J. Wright 1998 and W.W. Hager 1999) in finite dimensions to this broader setting.

Pricing European call options with interval-valued volatility and interest rate

We propose a novel approach to pricing European call options when both of the volatility of the underlying asset and interest are uncertain. In this approach, we formulate the option pricing problem with uncertain parameters as a partial-differential inequality constrained interval optimization problem. An interior penalty method is then developed for the numerical solution of the finite-dimensional optimization problem arising from the discretization of the continuous pricing problem by a finite difference scheme. A convergence theory for the penalty method is established. An algorithm based on Newton's iterative method is also proposed for solving the penalty equation. Numerical results are presented to demonstrate the effectiveness and usefulness of this approach and the numerical methods.

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.

Shape optimization for the Stokes system with threshold leak boundary conditions

This paper discusses the process of optimizing the shape of systems that are controlled by the Stokes flow with threshold leak boundary conditions. In the theoretical part it focuses on studying the stability of solutions to the state problem in relation to a specific set of domains. In order to facilitate computation, the slip term and impermeability condition are regulated. In the computational part, the optimized portion of the boundary is defined using Bézier polynomials, in order to create a finite dimensional optimization problem. The paper also includes numerical examples to demonstrate the computational efficiency of this approach.

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.

A robust optimal control framework for controlling aberrant RTK signaling pathways in esophageal cancer.

This study presents a new framework for obtaining personalized optimal treatment strategies targeting aberrant signaling pathways in esophageal cancer, such as the epidermal growth factor (EGF) and vascular endothelial growth factor (VEGF) signaling pathways. A new pharmacokinetic model is developed taking into account specific heterogeneities of these signaling mechanisms. The optimal therapies are designed to be obtained using a three step process. First, a finite-dimensional constrained optimization problem is solved to obtain the parameters of the pharmacokinetic model, using discrete patient data measurements. Next, a sensitivity analysis is carried out to determine which of the parameters are sensitive to the evolution of the variants of EGF receptors and VEGF receptors. Finally, a second optimal control problem is solved based on the sensitivity analysis results, using a modified pharmacokinetic model that incorporates two representative drugs Trastuzumab and Bevacizumab, targeting EGF and VEGF, respectively. Numerical results with the combination of the two drugs demonstrate the efficiency of the proposed framework.

Douglas–Rachford algorithm for control-constrained minimum-energy control problems

Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize the Douglas–Rachford (DR) algorithm to solve control-constrained minimum-energy optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the DR algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.

Sequential adaptive switching time optimization technique for maximum hands-off control problems

&lt;abstract&gt;&lt;p&gt;In this paper, we consider maximum hands-off control problem governed by a nonlinear dynamical system, where the maximum hands-off control constraint is characterized by an $ L^{0} $ norm. For this problem, we first approximate the $ L^{0} $ norm constraint by a $ L^{1} $ norm constraint. Then, the control parameterization together with sequential adaptive switching time optimization technique is proposed to approximate the optimal control problem by a sequence of finite-dimensional optimization problems. Furthermore, a smoothing technique is exploited to approximate the non-smooth maximum operator and an error analysis is investigated for this approximation. The gradients of the cost functional with respect to the decision variables in the approximate problem are derived. On the basis of these results, we develop a gradient-based optimization algorithm to solve the resulting optimization problem. Finally, an example is solved to demonstrate the effectiveness of the proposed algorithm.&lt;/p&gt;&lt;/abstract&gt;

On asymptotic convergence rate of random search

This paper presents general theoretical studies on asymptotic convergence rate (ACR) for finite dimensional optimization. Given the continuous problem function and discrete time stochastic optimization process, the ACR is the optimal constant for control of the asymptotic behaviour of the expected approximation errors. Under general assumptions, condition ACR<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$<1$$\\end{document} implies the linear behaviour of the expected time of hitting the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}- optimal sublevel set with ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+ $$\\end{document} and determines the upper bound for the convergence rate of the trajectories of the process. This paper provides general characterization of ACR and, in particular, shows that some algorithms cannot converge linearly fast for any nontrivial continuous optimization problem. The relation between asymptotic convergence rate in the objective space and asymptotic convergence rate in the search space is provided. Examples and numerical simulations with use of a (1+1) self-adaptive evolution strategy and other algorithms are presented.

Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

Kernel-Based Impulse Response Identification With Side-Information on Steady-State Gain

In this paper, we consider the problem of system identification when side-information is available on the steady-state gain (SSG) of the system. We formulate a general nonparametric identification method as an infinite-dimensional constrained convex program over the reproducing kernel Hilbert space (RKHS) of stable impulse responses. The objective function of this optimization problem is the empirical loss regularized with the norm of RKHS, and the constraint is considered for enforcing the integration of the SSG side-information. The proposed formulation addresses both the discrete-time and continuous-time cases. We show that this program has a unique solution obtained by solving an equivalent finite-dimensional convex optimization. This solution has a closed-form when the empirical loss and regularization functions are quadratic and exact side-information is considered. We perform extensive numerical comparisons to verify the efficiency of the proposed identification methodology.

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 $\ell _p (0&amp;lt;p&amp;lt;1)$ 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.

Трансмерная гибридизация эволюционных алгоритмов многокритериальной оптимизации управления

The paper presents a hybrid evolutionary algorithm for multicriteria synthesis of the optimal control law for a dynamic system based on the trans-dimensional search models. The developed trans-dimensional search model belongs to the class of sequential hybridization models of the preprocessor/processor type and implies the combined use of the evolutionary algorithms of finite-dimensional and infinite-dimensional multi-objective optimization implementing the stages of global approximate and local refinement search for the optimal solutions. A finite-dimensional model of the global multi-criteria optimization is implemented using an evolutionary algorithm of the multi-criteria optimization in regard to the polyhedral dominance cone. Introduction of the uncertainty intervals of the vector indicator components weight coefficients in constructing the dominance polyhedral cone makes it possible to reduce the Pareto set uncertainty by highlighting a subset of solutions in it that are having a higher degree of balancing values for various components of the efficiency vector indicator. Evolutionary algorithm of the infinite-dimensional multi-objective optimization is a generalization of the Zoytendijk’s methods of possible directions for the class of infinite-dimensional multi-objective optimization problems and is used at the search stage. The paper provides results of a comparative analysis of various hybrid trans-dimensional models efficiency in the evolutionary search on the example of solving the problem of multicriteria synthesis of the optimal law for a bioreactor program control. Results of the computational experiments show that trans-dimensional hybridization of the evolutionary algorithms for the finite-dimensional and infinite-dimensional multicriteria control optimization provides a synergistic effect. This effect is expressed in significant increase in the accuracy of solving the problem of multicriteria control optimization in comparison with the known hybrid metaheuristic control optimization algorithms making it possible to resolve contradiction between the finite-dimensional global search model and the infinite-dimensional initial problem formulation

Robust optimal control of nonlinear fractional systems

Parameter uncertainty is inevitably found in almost every system and, if is ignored, it could jeopardize the effectiveness of control method. Motivated by this issue, we consider a robust optimal control problem governed by a nonlinear fractional system with uncertain parameters, where the system sensitivity with respect to the uncertain parameters is explicitly included in the cost functional. For this problem, we first prove that the system sensitivity can be expressed as the solution of an auxiliary fractional system. Then, we propose a numerical scheme for discretizing both the original and auxiliary fractional systems, resulting in a finite-dimensional optimization problem. Furthermore, a numerical solution algorithm based on gradients of the cost functional is developed for the resulting optimization problem. Finally, numerical results for two example problems are given to demonstrate the validity and applicability of the proposed algorithm.

A computationally efficient sequential convex programming using Chebyshev collocation method

This paper aims to develop a computationally efficient sequential convex programming algorithm for a class of nonlinear systems. A Chebyshev collocation discretization (CCD) technique is proposed that transforms the continuous convex optimization problem into a finite-dimensional discrete optimization problem. The CCD technique uses Chebyshev polynomials to construct a closed-form approximate solution to the convex dynamic equation. Moreover, it establishes a linear mapping between the control inputs and the system states. This enables the constraints of the dynamics equations to be externalized from the optimization problem and computed using efficient linear equation solving algorithms. On the one hand, this strategy significantly reduces the number of constraint equations and optimization variables, improving the speed of the optimizer. On the other hand, it preserves the path constraints on the system states in the optimization problem. A numerical simulation example of the perching maneuver for a fixed-wing unmanned aerial vehicle is presented to validate the efficiency of the algorithm. The result shows significant improvements in both speed and accuracy compared to the Euler discretization, and greatly improves the solution speed while maintaining almost the same solution accuracy compared to the Gauss pseudospectral optimization method.

Optimal Noise-Boosted Estimator Design Via Adaptive Stochastic Resonance

In signal estimation, an optimal estimator is frequently unachievable because its closed form may not be analytically tractable or is too complex to implement. Alternatively, one can turn to suboptimal yet easily implementable estimators for practical signal estimation tasks. In this paper, an optimal noise-boosted estimator is designed and the adaptive stochastic resonance method is implemented to simultaneously exploit the beneficial role of the injected noise as well as the learning ability of the estimator parameter. Aiming to effectively improve the estimation performance, we use the kernel function method to find an approximate solution for the probability density function (PDF) of the optimal injected noise. During this process, the noise PDF and the estimator parameter establish a finite-dimensional non-convex optimization space for maximizing the estimation performance, which is adaptively searched by the sequential quadratic programming (SQP) algorithm at each iteration. Two representative estimation problems are explored. The obtained results demonstrate that this adaptive stochastic resonance method can improve the performance of the suboptimal estimators and bring it very close to that of the optimal estimator.