Related Optimization Problems Research Articles

Derivative-free bound-constrained optimization for solving structured problems with surrogate models

We propose and analyze a model-based derivative-free (DFO) algorithm for solving bound-constrained optimization problems where the objective function is the composition of a smooth function and a vector of black-box functions. We assume that the black-box functions are smooth and the evaluation of them is the computational bottleneck of the algorithm. The distinguishing feature of our algorithm is the use of approximate function values at interpolation points which can be obtained by an application-specific surrogate model that is cheap to evaluate. As an example, we consider the situation in which a sequence of related optimization problems is solved and present a regression-based approximation scheme that uses function values that were evaluated when solving prior problem instances. In addition, we propose and analyze a new algorithm for obtaining interpolation points that handles unrelaxable bound constraints. Our numerical results show that our algorithm outperforms a state-of-the-art DFO algorithm for solving a least-squares problem from a chemical engineering application when a history of black-box function evaluations is available.

Optimal resilience-based restoration of a system subject to recurrent dependent hazards

Many engineered systems are faced with multiple natural and manmade hazards during operation. Since such hazards may cause system disruptions (e.g., catastrophic failure and performance deterioration), timely and effective restoration needs to be conducted to minimize the negative impacts of the hazards. In practice, these hazards are recurrent and often interact dynamically with each other. This causes a major challenge in assessing the performance of the systems and determining the optimal restoration policy from a holistic perspective. In this paper, we model the availability and resilience of a system subject to recurrent and dependent hazards and develop the optimal restoration policies for decision makers with different risk preferences. Particularly, we first model the random occurrence times and severities of a set of hazards whose occurrence frequencies and severities are dynamically dependent. Next, we derive the analytical expressions for the system's availability metrics. Third, we investigate the optimal restoration policies for both risk-neutral (RN) and risk-averse (RA) decision makers by solving two related optimization problems. A case study on the pipelines in the Mid-Atlantic area of the U.S. is provided to illustrate the practical values of the proposed availability assessment and resilience optimization methods.

GSO-Net: Grid Surface Optimization via Learning Geometric Constraints

In the context of surface representations, we find a natural structural similarity between grid surface and image data. Motivated by this inspiration, we propose a novel approach: encoding grid surfaces as geometric images and using image processing methods to address surface optimization-related problems. As a result, we have created the first dataset for grid surface optimization and devised a learning-based grid surface optimization network specifically tailored to geometric images, addressing the surface optimization problem through a data-driven learning of geometric constraints paradigm. We conduct extensive experiments on developable surface optimization, surface flattening, and surface denoising tasks using the designed network and datasets. The results demonstrate that our proposed method not only addresses the surface optimization problem better than traditional numerical optimization methods, especially for complex surfaces, but also boosts the optimization speed by multiple orders of magnitude. This pioneering study successfully applies deep learning methods to the field of surface optimization and provides a new solution paradigm for similar tasks, which will provide inspiration and guidance for future developments in the field of discrete surface optimization. The code and dataset are available at https://github.com/chaoyunwang/GSO-Net.

Automated importance sampling via optimal control for stochastic reaction networks: A Markovian projection–based approach

We propose a novel alternative approach to our previous work (Ben Hammouda et al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare event probabilities for stochastic reaction networks (SRNs). In the same spirit of Ben Hammouda et al. (2023), an efficient path-dependent measure change is derived based on a connection between determining optimal importance sampling (IS) parameters within a class of probability measures and a stochastic optimal control formulation, corresponding to solving a variance minimization problem. In this work, we propose a novel approach to address the encountered curse of dimensionality by mapping the problem to a significantly lower-dimensional space via a Markovian projection (MP) idea. The output of this model reduction technique is a low-dimensional SRN (potentially even one dimensional) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained by solving a related optimization problem via a discrete L2 regression. By solving the resulting projected Hamilton–Jacobi–Bellman (HJB) equations for the reduced-dimensional SRN, we obtain projected IS parameters, which are then mapped back to the original full-dimensional SRN system, resulting in an efficient IS-MC estimator for rare events probabilities of the full-dimensional SRN. Our analysis and numerical experiments reveal that the proposed MP-HJB-IS approach substantially reduces the MC estimator variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.

Whittle index approach to multiserver scheduling with impatient customers and DHR service times

AbstractWe consider the optimal scheduling problem in a multiserver queue with impatient customers belonging to multiple classes. We assume that each customer has a random abandonment time, after which the customer leaves the system if its service has not been completed before that. In addition, we assume that the scheduler is not able to anticipate the expiration of the abandonment times but only knows their distributions and how long each customer has been in the system. Many papers consider this scheduling problem under Poisson arrivals and linear holding costs assuming further that both the service times and the abandonment times have exponential distributions. Even with these additional assumptions, the exact solution is known only in very few special cases. To tackle this tricky problem, we apply the Whittle index approach. Unlike the earlier papers, which were restricted to exponential service times, we allow the service time distributions for which the hazard rate is decreasing. The Whittle index approach is applied to the discrete-time multiserver queueing problem with discounted costs. As our main theoretical result, we prove that the related relaxed optimization problem is indexable and derive the corresponding Whittle index explicitly. Based on this discrete-time result, we develop a reasonable heuristic for the original continuous-time multiserver scheduling problem. The performance of the resulting policy is evaluated in the M/G/M setup by numerical simulations, which demonstrate that it, indeed, gives better performance than the other policies included in the comparison.

On Monotonic Radial Basis Function Networks.

This article deals with monotonicity conditions for radial basis function (RBF) networks. Two architectures of RBF networks are considered-1) unnormalized network with a local character of the basis function and 2) a normalized network where the value of RBF is taken relatively with respect to the others. Different approaches are followed for each of them. For the former, monotonicity is enforced in prescribed points whereas for the latter sufficient monotonicity conditions are formulated. In both cases, the monotonicity conditions are expressed as linear constraints on the network weights that enable efficient solving of the related optimization problems. Many illustrative examples are presented to show the advantages of incorporating prior information in the form of monotonicity.

Robust GEPSVM Classifier: An Efficient Iterative Optimization Framework

The proximal support vector machine via generalized eigenvalues (GEPSVM) is a well-known pattern classification method. GEPSVM, however, is prone to outliers due to its use of the squared L2-norm distance criterion. A robust GEPSVM version is proposed to tackle this problem using L1-norm distance optimization technique (GEPSVML1). As optimizing a GEPSVML1 with L1-norm terms can be challenging, we have developed an iterative algorithm to address the L1-norm ratio problem associated with GEPSVML1. Furthermore, an efficient iterative optimization framework has been developed to conveniently address related optimization problems. The research contribution of this paper lies in providing a theoretical analysis of the algorithm’s convergence. Besides, we find that the L1-norm distance-based methods in real-world applications, especially for handling samples with outliers, sometimes provides an unsatisfactory recognition result. Thus, a generalized version of GEPSVML1 is proposed. The L1-norm distance is replaced with a Lp-norm distance in GEPSVML1 (GEPSVMLp). It is the robust counterpart of GEPSVML1 and GEPSVM. It is worth noting that we fine-tune GEPSVMLp’s parameters to strike a balance between training time and classification accuracy, an especially crucial step for larger datasets. Our experiments indicate that the proposed GEPSVMLp is more efficient and robust than the competitors in numerous experimental settings. Overall, our work demonstrates the importance of developing robust pattern classification methods in the presence of outliers and provides a practical solution for handling such cases in real-world applications.

An adaptive large neighborhood search for the order picking process: the case of a retail distribution company in Italy

Order picking is one of the most important activities in warehouse management. By optimising the picking process, warehouse operations can be managed efficiently in terms of both time and logistics costs. In this work we apply operations research techniques to support the order picking process of an Italian retail distribution company. The picking process poses several challenges from an optimisation point of view, and the related optimization problems are very complex. Therefore, we define an Adaptive Large Neighbourhood Search (ALNS) algorithm that heuristically solves the problem. The proposed metaheuristic is tested on a set of 101 real orders processed by the company within one day. The computational experiments show that the ALNS shows good performances in terms of effectiveness, compared to the optimal solution, as well as allows to implement a better organisation of the order picking process than the one currently adopted by the company.

Finite time convergent recurrent neural network for variational inequality problems subject to equality constraints

This paper aims to present a new class of recurrent neural networks based on projection operators for solving variational inequalities and related optimization problems subject to linear equalities and bound constraints. In the proposed approach, the neural network structure is based on the Karush–Kuhn–Tucker (KKT) optimality conditions. Instead of commonly used activation functions, the KKT multipliers are considered control inputs and implemented with finite time stabilizing terms based on unit control. The output variables of the neural network are proven to be stable in the sense of Lyapunov and convergent to optimal solutions in finite time. In addition, a methodology based on the particle swarm optimization (PSO) algorithm is presented for the optimal selection of network design parameters. The main advantage of the proposed neural network is its fixed number of parameters regardless of problem dimension, enabling the network to be easily scalable from a lower to a higher dimension. Finally, simulation results of numerical examples are provided to illustrate the effectiveness and performance of the proposed neural network.

Optimal management of DB pension fund under both underfunded and overfunded cases

This paper investigates the optimal management of an aggregated defined benefit pension plan in a stochastic environment. The interest rate follows the Ornstein-Uhlenbeck model, the benefits follow the geometric Brownian motion while the contribution rate is determined by the spread method of fund amortization. The pension manager invests in the financial market with three assets: cash, a zero-coupon bond and a stock. Regardless of the initial status of the plan, we suppose that the pension fund may become underfunded or overfunded in the planning horizon. The optimization goal of the manager is to maximize the expected utility in the overfunded region minus the weighted solvency risk in the underfunded region. By introducing an auxiliary process and related equivalent optimization problems and using the martingale method, the optimal wealth process, optimal portfolio and efficient frontier are obtained under four cases (high tolerance towards solvency risk, low tolerance towards solvency risk, a specific lower bound, and high lower bound). Moreover, we also obtain the probabilities that the optimal terminal wealth falls in the overfunded and underfunded regions. At last, we present numerical analyzes to illustrate the manager's economic behaviors.

Interval valued inventory model for deterioration, carbon emissions and selling price dependent demand considering buy now and pay later facility

Uncertainty refers to a lack of precise knowledge or information about a particular event, situation, or outcome. It is an inherent characteristic of many real-world phenomena and is often associated with the presence of risk or ambiguity. Uncertainty can arise due to various factors, such as incomplete information, unpredictability, complexity, or variability of a system or process. In many cases, uncertainty can lead to difficulties in decision-making and pose challenges in planning, forecasting, and managing risks. There are different types of uncertainty, including epistemic uncertainty, which arises from the inherent randomness or variability of a system or process. This study encompasses the inclusion of all inventory parameters as uncertainty parameters, which are expressed in the form of intervals. Furthermore, demand for the product is taken into account in interval form. The conversion of the differential equation into an interval differential equation is carried out in order to account for the variability in demand within a certain interval. The utilisation of the centre-radius technique leads to the derivation of the related optimisation problem under a ‘buy now, pay later’ (BNPL) payment scheme. In order to demonstrate the concept, a numerical case is selected and solved with the MATHEMATICA software and interval order relations. In the end, a comprehensive sensitivity analysis has been conducted to derive insightful conclusions pertaining to this study.

POLYNOMIAL UTILITY

We approximate the utility function by polynomial series and solve the related dynamic portfolio optimization problems. We study the quality of the Taylor and Bernstein series approximation in response to the points and degrees of the expansions and generalize from earlier expansions applied to portfolio optimization. The issue of time inconsistency, arising from a dynamically adapted center of the expansion, is approached by equilibrium theory. We present new ways of constructing polynomial utility functions and study their pitfalls and potentials. In the numerical study, we focus on two specific utility functions: For power utility, access to the optimal portfolio allows for a complete illustration of the approximations; for the S-shaped utility function of prospect theory, the use of equilibrium theory allows for approximating the solution to the (obviously interesting but yet unsolved) case of current wealth as a dynamic reference point.

Reverse Auction-Based Computation Offloading and Resource Allocation in Mobile Cloud-Edge Computing

This paper proposes a novel Reverse Auction-based Computation Offloading and Resource Allocation Mechanism, named RACORAM for the mobile Cloud-Edge computing. The basic idea is that the Cloud Service Center (CSC) recruits edge server owners to replace it to accommodate offloaded computation from nearby resource-constraint Mobile Devices (MDs). In RACORAM, the reverse auction is used to stimulate edge server owners to participate in the offloading process, and the reverse auction-based computation offloading and resource allocation problem is formulated as a Mixed Integer Nonlinear Programming (MINLP) problem, aiming to minimize the cost of the CSC. The original problem is decomposed into an equivalent master problem and subproblem, and low-complexity algorithms are proposed to solve the related optimization problems. Specifically, a Constrained Gradient Descent Allocation Method (CGDAM) is first proposed to determine the computation resource allocation strategy, and then a Greedy Randomized Adaptive Search Procedure based Winning Bid Scheduling Method (GWBSM) is proposed to determine the computation offloading strategy. Meanwhile, the CSC’s payment determination for the winning edge server owners is also presented. Simulations are conducted to evaluate the performance of RACORAM, and the results show that RACORAM is very close to the optimal method with significantly reduced computational complexity, and greatly outperforms the other baseline methods in terms of the CSC’s cost under different scenarios.

On polynomially solvable constrained input selections for fixed and switched linear structured systems

This paper investigates two related optimal input selection problems for fixed (non-switched) and switched structured (or structural) systems. More precisely, we consider selecting the minimum cost of inputs from a prior set of inputs, and selecting the inputs of the smallest possible cost with a bound on their cardinality, all to ensure system structural controllability. Those problems have attracted much attention recently; unfortunately, they are NP-hard in general. In this paper, it is found that, if the input structure satisfies certain ’regularizations’, which are characterized by the proposed restricted total unimodularity notion, those problems can be solvable in polynomial time via linear programming (LP) relaxations. Particularly, the obtained characterizations depend only on the incidence matrix relating the inputs and the source strongly connected components (SCC) of the system structure, irrespective of how the inputs actuate states within the same SCC. They cover all the currently known polynomially solvable cases (such as the dedicated input case), and contain many new cases unexplored in the past, among which the source-SCC separated input (SSSI) constraint is highlighted. Further, these results are extended to switched systems, and a polynomially solvable condition, namely the joint SSSI constraint, is obtained that does not require each of the subsystems to satisfy the SSSI constraint. We achieve these by first formulating those problems as equivalent integer linear programmings (ILPs), and then proving the total unimodularity of the corresponding constraint matrices. We also study solutions obtained via LP-relaxation and LP-rounding in the general case, resulting in some lower and upper bounds. Several examples are given to illustrate the obtained theoretical results.

Novel Multistep Implicit Iterative Methods for Solving Common Solution Problems with Asymptotically Demicontractive Operators and Applications

It is very meaningful and challenging to efficiently seek common solutions to operator systems (CSOSs), which are widespread in pure and applied mathematics, as well as some closely related optimization problems. The purpose of this paper is to introduce a novel class of multistep implicit iterative algorithms (MSIIAs) for solving general CSOSs. By using Xu’s lemma and Maingé’s fundamental and important results, we first obtain strong convergence theorems for both one-step and multistep implicit iterative schemes for CSOSs, involving asymptotically demicontractive operators. Finally, for the applications and profits of the main results presented in this paper, we give two numerical examples and present an iterative approximation to solve the general common solution to the variational inequalities and operator equations.

Some New Classes of Harmonic Hemivariational Inequalities

Some new classes of harmonic hemivariational inequalities are introduced and investigated in this paper. It has been shown that the optimality conditions of the sum of two harmonic convex functions can be characterized by the harmonic hemivariational inequalities. Several special cases such as harmonic complementarity problems and related harmonic problems are discussed. The auxiliary principle technique is applied to suggest and analyze some iterative schemes for harmonic hemivariational inequalities. We prove the convergence of these iterative methods under some weak conditions. Our method of proof of the convergence criteria is simple compared to other techniques. Results obtained in this paper continue to hold for new and known classes of harmonic variational inequalities and related optimization problems. The ideas and techniques of this paper may inspire further research in various branches of pure and applied sciences.

Cycle selections

We introduce the following cycle selection problem which is motivated by an application to kidney exchange problems. Given a directed graph G = ( V , A ) , a cycle selection is a subset of arcs B ⊆ A forming a union of directed cycles. A related optimization problem, the Maximum Weighted Cycle Selection problem can be defined as follows: given a weight w i , j ∈ R for all arcs ( i , j ) ∈ A , find a cycle selection B which maximizes w ( B ) . We prove that this problem is strongly NP-hard. Next, we focus on cycle selections in complete directed graphs. We provide several ILP formulations of the problem: an arc formulation featuring an exponential number of constraints which can be separated in polynomial time, four extended compact formulations, and an extended non compact formulation. We investigate the relative strength of these formulations. We concentrate on the arc formulation and on the description of the associated cycle selection polytope. We prove that this polytope is full-dimensional, and that all the inequalities used in the arc formulation are facet-defining. Furthermore, we describe three new classes of facet-defining inequalities and a class of valid inequalities. We also consider the consequences of including additional constraints on the cardinality of a selection or on the length of the associated cycles.

Sparse solution of least-squares twin multi-class support vector machine using [formula omitted] and [formula omitted]-norm for classification and feature selection

In the realm of multi-class classification, the twin K-class support vector classification (Twin-KSVC) generates ternary outputs {−1,0,+1} by evaluating all training data in a “1-versus-1-versus-rest” structure. Recently, inspired by the least-squares version of Twin-KSVC and Twin-KSVC, a new multi-class classifier called improvements on least-squares twin multi-class classification support vector machine (ILSTKSVC) has been proposed. In this method, the concept of structural risk minimization is achieved by incorporating a regularization term in addition to the minimization of empirical risk. Twin-KSVC and its improvements have an influence on classification accuracy. Another aspect influencing classification accuracy is feature selection, which is a critical stage in machine learning, especially when working with high-dimensional datasets. However, most prior studies have not addressed this crucial aspect. In this study, motivated by ILSTKSVC and the cardinality-constrained optimization problem, we propose ℓp-norm least-squares twin multi-class support vector machine (PLSTKSVC) with 0<p<1 to perform classification and feature selection at the same time. The technique employed to solve the optimization problems associated with PLSTKSVC is user-friendly, as it involves solving systems of linear equations to obtain an approximate solution for the proposed model. Under certain assumptions, we investigate the properties of the optimum solutions to the related optimization problems. Several real-world datasets were tested using the suggested method. According to the results of our experiments, the proposed method outperforms all current strategies in most datasets in terms of classification accuracy while also reducing the number of features.

Combination of electromagnetic field and harris hawks optimization algorithms with optimization to optimization structure and its application for optimum power flow

ABSTRACT Electromagnetic Field Optimization (EFO) and Harris Hawk Optimization (HHO) algorithms are combined with the optimization to optimization (OtoO) approach, and the EFO-HHO algorithm pair is presented in this study. EFO method was used as the essential algorithm and HHO method was used as the auxiliary algorithm according to the OtoO structure. The constant parameters (R_rate, Ps_rate, P_field, N_field) of the EFO algorithm that affect the optimization performance are optimized with the HHO optimization algorithm for the related optimization problem. The proposed method was tested on 10 different benchmark functions according to different dimensional (30, 50100). The EFO-HHO algorithm pair can produce better results than the existing literature, especially in cases of increased dimension with the proposed approach. In addition to these, the OPF problem was tested on the IEEE 30 test bus system for the engineering application of the proposed method. The results are compared with the existing literature results. As it can be seen from the results, it has been shown on the real engineering problem that the optimization performance can be increased with the OtoO approach without changing the basic philosophy of the EFO algorithm.

Error Bounds and Merit Functions for Exponentially General Variational Inequalities

In this paper, some new classes of classes of exponentially general variational inequalities are introduced. It is shown that the odd-order and nonsymmetric exponentially boundary value problems can be studied in the framework of exponentially general variational inequalities. We consider some classes of merit functions for exponentially general variational inequalities. Using these functions, we derive error bounds for the solution of exponentially general variational inequalities under some mild conditions. Since the exponentially general variational inequalities include general variational inequalities, quasi-variational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. Results obtained in this paper represent a refinement of previously known results for several classes of variational inequalities and related optimization problems.