Stochastic Objective Function Research Articles

Probabilistic branch and bound considering stochastic constraints

In this paper, we investigate a simulation optimization problem that poses challenges due to (i) the inability to evaluate the objective and multiple constraint functions analytically; instead, we rely on stochastic simulation to estimate them, and (ii) a discrete and potentially vast solution space. Rather than providing a single optimal solution, our aim is to identify a set of near-optimal solutions within a specific quantile, such as the top 10%. Our investigation covers two different problem settings or frameworks. The first framework is focused solely on a stochastic objective function, disregarding any stochastic constraints. In this context, we propose employing a probabilistic branch-and-bound algorithm to discover a level set of solutions. Alternatively, the second framework involves stochastic constraints. To address such stochastically constrained problems, we utilize a penalty function methodology in conjunction with a probabilistic branch-and-bound algorithm. Furthermore, we establish a convergence analysis of both algorithms to demonstrate their asymptotic validity and highlight their theoretical properties and behavior. Our experimental results provide evidence of the efficiency of our proposed algorithms, showing that they outperform existing approaches in the field of simulation optimization.

Online distributed nonconvex optimization with stochastic objective functions: High probability bound analysis of dynamic regrets

In this paper, the problem of online distributed optimization with stochastic and nonconvex objective functions is studied by employing a multi-agent system. When making decisions, each agent only has access to a noisy gradient of its own objective function in the previous time and can only communicate with its immediate neighbors via a time-varying digraph. To handle this problem, an online distributed stochastic projection-free algorithm is proposed. Of particular interest is that the dynamic regrets are employed to measure the performance of the online algorithm. Existing works on online distributed algorithms involving stochastic gradients only provide the sublinearity results of regrets in expectation. Different from them, we study the high probability bounds of dynamic regrets, i.e., the sublinear bounds of dynamic regrets are characterized by the natural logarithm of the failure probability’s inverse. Under mild assumptions on the graph and objective functions, we prove that if the variations in both the objective function sequence and its gradient sequence grow within a certain rate, then the high probability bounds of the dynamic regrets grow sublinearly. Finally, a simulation example is carried out to demonstrate the effectiveness of our theoretical results.

Convergence property of Nesterov‐accelerated adaptive moment estimation with safety helmet detection and classification in smart industry application

We propose a technique for first‐order gradient‐based optimization of stochastic objective functions called Nesterov‐accelerated adaptive moment assessment, which makes use of dynamic evaluations of lower‐order moments. The adaptive moment assessment and the Nesterov acceleration gradient are combined. Consequently, it has perks, and this technique is convenient to use, numerically economical, memory‐light, and very well‐suited for challenges with massive amounts of information and characteristics. Additionally, we investigate the algorithm's convergence characteristics and propose a conservative constraint on the convergence rate. Finally, we employ this technique for the detection and classification of safety helmets.

3D mineral prospectivity modeling in the Sanshandao goldfield, China using the convolutional neural network with attention mechanism

Mineralization distribution is spatially heterogeneous and jointly controlled by multiple ore-controlling factors. Given this ubiquitous feature, we integrated the attention mechanism of the Convolutional Block Attention Module (CBAM), which contains both channel and spatial attention modules, into a convolutional neural network (CNN) to develop a more robust 3D mineral prospectivity modeling (MPM) approach. A network interpretive algorithm (DeepLIFT) was conducted to the CNN deconstruction to evaluate whether the key ore-controlling information is read. The Sanshandao goldfield in China, with significant fault controls on gold mineralization distribution, was selected as a case study to verify the proposed method. In this case, we generated the 5-channel images by projecting five ore-related fault features to feed into CNN. The optimal prospectivity model of CNN + CBAM was subsequently yielded by a gradient-based optimization of stochastic objective functions and an optimized regularization cross-entropy loss function. By decomposing the constructed model and the vanilla CNN model, we observed the CNN + CBAM has lower feature scores of fault distance and larger scores of fault geometry (e.g., dip and undulation) at Xiling, Haiyu, and Xinli than CNN. These segments occur significant fault strike or dip changes that have influenced mineralization spatial continuity, which is well consistent with the expert knowledge of metallogeny in the Sanshandao goldfield. This means that the CBAM has induced the CNN to focus on key ore-controlling features from complex exploration data, which is likely to improve the model’s prediction ability. The prospectivity results of CNN + CBAM and other comparison shallow machine learning methods (e.g., random forest) revealed that the CNN + CBAM model produced more reliable results, as evidenced by (1) being highly consistent with known mineralization, (2) having the highest AUC values and success rate, and (3) accurately predicting deep voxels that have been explored by drill holes. Thus, the proposed CNN + CBAM method, with a remarkable capacity for emphasizing critical ore-associated features, is recommended for application in deep 3D MPM. Six possible mineralization sites in the Sanshandao field were identified based on the CNN + CBAM prospective data, which may aid future gold exploration.

The Stochastic Robustness of Nominal and Stochastic Model Predictive Control

In this work, we establish and compare the stochastic and deterministic robustness properties achieved by nominal model predictive control (MPC), stochastic MPC (SMPC), and a proposed constraint-tightened MPC (CMPC) formulation, which represents an idealized version of tube-based MPC. We consider three definitions of robustness for nonlinear systems and bounded disturbances: robust asymptotic stability (RAS), robust asymptotic stability in expectation (RASiE), and RASiE w.r.t. the stage cost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell (\cdot)$</tex-math></inline-formula> used in these MPC formulations ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> -RASiE). Via input-to-state stability (ISS) and stochastic ISS (SISS) Lyapunov functions, we establish that MPC, subject to sufficiently small disturbances, and CMPC ensure all three definitions of robustness without a stochastic objective function. While SMPC also ensures RASiE and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell$</tex-math></inline-formula> -RASiE, SMPC does not guarantee RAS for nonlinear systems. Through a few simple examples, we illustrate the implications of these results and demonstrate that, depending on the definition of robustness considered, SMPC is not necessarily more robust than nominal MPC even if the disturbance model is exact.

Modifications of generalized response surface methodology for constrained stochastic optimization problems

To optimize problems with a stochastic objective function and some stochastic or deterministic constraints, the generalized response surface methodology (GRSM) is used. This study presents a novel GRSM that is fast and accurate enough to solve constrained stochastic optimization problems containing expensive simulations or too costly real experiments. Basic GRSM is a kind of interior point method which moves toward the optimum from only inside of the feasible region. However, the proposed approach can move toward the optimum from inside or outside the feasible region. In addition, basic GRSM is not always reliable and fails in problems including complicated stochastic functions; we improved the GRSM to enable it to make reliable solutions to complicated problems. We solve two examples: 1- a toy problem with two stochastic constraints, and 2- an optimization problem of inventory system with one stochastic constraint. The obtained results indicate the optimization method proposed in this study is much more efficient than OPT-QUEST and classic GRSM.

Parametric level-set inverse problems with stochastic background estimation

We study parametric shape reconstruction inverse problems in which the object of interest is embedded in a heterogeneous background medium that is known only approximately. We model the background medium as a Gaussian random field and pose shape reconstruction as a stochastic programming problem in which we seek to minimize the expected value, with respect to the background field, of a stochastic objective function. We develop a computationally efficient algorithm based on the sample average approximation that reduces the effect of uncertainty in the background medium on shape recovery. We demonstrate that by using accelerated stochastic gradient descent, we can apply our method to large-scale problems. The capabilities of our method are demonstrated on a simple two-dimensional model problem and in a more demanding application to a three-dimensional inverse conductivity problem in geophysical imaging.

Multi-objective Allocation Optimization of Soil Conservation Measures Under Data Uncertainty.

Many regions worldwide face soil loss rates that endanger future food supply. Constructing soil and water conservation measures reduces soil loss but comes with high labor costs. Multi-objective optimization allows considering both soil loss rates and labor costs, however, required spatial data contain uncertainties. Spatial data uncertainty has not been considered for allocating soil and water conservation measures. We propose a multi-objective genetic algorithm with stochastic objective functions considering uncertain soil and precipitation variables to overcome this gap. We conducted the study in three rural areas in Ethiopia. Uncertain precipitation and soil properties propagate to uncertain soil loss rates with values that range up to 14%. Uncertain soil properties complicate the classification into stable or unstable soil, which affects estimating labor requirements. The obtained labor requirement estimates range up to 15 labor days per hectare. Upon further analysis of common patterns in optimal solutions, we conclude that the results can help determine optimal final and intermediate construction stages and that the modeling and the consideration of spatial data uncertainty play a crucial role in identifying optimal solutions.

Profit Maximization in ADN Through Voltage Control and DR Management With Networked Community Micro-Grids

Large-scale installations of utility and customer's owned distributed technologies at low and medium voltage networks have drawn the distribution system operator's attention towards enhancing energy efficiency and economics of distribution grids. It can be achieved by deploying suitable energy-saving programs and unlocking the benefits of networked community microgrids (N-CMGs). In this context, the paper introduces an integrated methodology by enabling the volt/voltage optimization and demand response management scheme in the presence of N-CMGs that leverage distributed energy resources, flexible loads, and a smart energy management system. The stochastic objective function has been formulated regarding net profit value maximization in the presence of N-CMG assets. Further, efficient cascaded droop (Volt/VAR, Volt/Watt) control algorithms have been developed for real-time control of N-CMGs operations under unexpected transient conditions. The simulated outcomes authenticate that the proposed integrated method maximized the net profit values in terms of reduced purchased electricity cost and improved energy efficiency. Moreover, the developed real-time cascaded droop controller for N-CMGs successfully handles the unprecedented event's impact on system security during real-time operation.

Distributed stochastic nonsmooth nonconvex optimization

Distributed consensus optimization has received considerable attention in recent years and several distributed consensus-based algorithms have been proposed for (nonsmooth) convex and (smooth) nonconvex objective functions. However, the behavior of these distributed algorithms on nonconvex, nonsmooth and stochastic objective functions is not understood. Such class of functions and distributed setting are motivated by several applications, including problems in machine learning and signal processing. This paper presents the first convergence analysis of the decentralized stochastic subgradient method for such classes of problems, over networks modeled as undirected, fixed, graphs.

An Adaptive Gradient Method with Energy and Momentum

We introduce a novel algorithm for gradient-based optimization of stochastic objective functions. The method may be seen as a variant of SGD with momentum equipped with an adaptive learning rate automatically adjusted by an 'energy' variable. The method is simple to implement, computationally efficient, and well suited for large-scale machine learning problems. The method exhibits unconditional energy stability for any size of the base learning rate. We provide a regret bound on the convergence rate under the online convex optimization framework. We also establish the energy-dependent convergence rate of the algorithm to a stationary point in the stochastic non-convex setting. In addition, a sufficient condition is provided to guarantee a positive lower threshold for the energy variable. Our experiments demonstrate that the algorithm converges fast while generalizing better than or as well as SGD with momentum in training deep neural networks, and compares also favorably to Adam.

Stochastic Learning Approach for Binary Optimization: Application to Bayesian Optimal Design of Experiments

We present a novel stochastic approach to binary optimization for optimal experimental design (OED) for Bayesian inverse problems governed by mathematical models such as partial differential equations. The OED utility function, namely, the regularized optimality criterion, is cast into a stochastic objective function in the form of an expectation over a multivariate Bernoulli distribution. The probabilistic objective is then solved by using a stochastic optimization routine to find an optimal observational policy. The proposed approach is analyzed from an optimization perspective and also from a machine learning perspective with correspondence to policy gradient reinforcement learning. The approach is demonstrated numerically by using an idealized two-dimensional Bayesian linear inverse problem, and validated by extensive numerical experiments carried out for sensor placement in a parameter identification setup.

Multiple Electricity Markets Competitiveness Undergoing Symmetric and Asymmetric Renewables Development Policies

This paper scrutinizes the impact of different renewable energy sources (RES) development policies on competitiveness within multiple electricity markets (MEMs). Also, the variation in market power indices by increasing the integration of the markets undergoing symmetric and asymmetric RES development policies is investigated. To do so, several stochastic mixed-integer non-linear programming objective functions are used in the agent-based simulation framework to model the power plants’ behavior and markets. The case study shows in the low RES penetrated markets, one can say the more integration level of the markets, the lower potential of exercising market power. The reciprocal judgment is true for a high RES penetrated market. Also, large asymmetry in RES development between markets within MEMs may bring about market power problem for a high RES penetrated market. Unlike the asymmetric RES development policies, adopting homogeneous policies in RES development within MEMs reduces the market power potential in all markets and this potential decreases with the increase in the integration of the markets.

Itao: A New Iterative Thresholding Algorithm Based Optimizer for Deep Neural Networks

In this paper, we propose a new iterative thresholding algorithm based optimizer (Itao) for deep neural networks. It is a first-order gradient-based algorithm with Tikhonov regularization for stochastic objective functions. It is fast and straightforward to implement. It acts on the parameters and their gradients, with respect to the objective function, in only one step in the backpropagation system when training a neural network. This reduces the learning time and makes it well suited for neural networks with large parameters and/or large datasets. We have experimented this algorithm on several types of loss functions such as mean squared error, mean absolute error and categorical crossentropy. Different types of models such as regression and classification are studied. The robustness of this optimizer against the noisy labels is also verified. Many of the empirical results of conducted experiments in this study show that our optimizer works well in practice. It can outperform other state-of-the-art optimizers in terms of accuracy or at least give the same results in addition to the reduction of learning time.

Integration of Ordinal Optimization with Ant Lion Optimization for Solving the Computationally Expensive Simulation Optimization Problems

The optimization of several practical large-scale engineering systems is computationally expensive. The computationally expensive simulation optimization problems (CESOP) are concerned about the limited budget being effectively allocated to meet a stochastic objective function which required running computationally expensive simulation. Although computing devices continue to increase in power, the complexity of evaluating a solution continues to keep pace. Ordinal optimization (OO) is developed as an efficient framework for solving CESOP. In this work, a heuristic algorithm integrating ordinal optimization with ant lion optimization (OALO) is proposed to solve the CESOP within a short period of time. The OALO algorithm comprises three parts: approximation model, global exploration, and local exploitation. Firstly, the multivariate adaptive regression splines (MARS) is adopted as a fitness estimation of a design. Next, a reformed ant lion optimization (RALO) is proposed to find N exceptional designs from the solution space. Finally, a ranking and selection procedure is used to decide a quasi-optimal design from the N exceptional designs. The OALO algorithm is applied to optimal queuing design in a communication system, which is formulated as a CESOP. The OALO algorithm is compared with three competing approaches. Test results reveal that the OALO algorithm identifies solutions with better solution quality and better computing efficiency than three competing algorithms.

Optimized Throughput in Covert Millimeter-Wave UAV Communications With Beam Sweeping

In this letter, we consider a unmanned-aerial-vehicle (UAV)-aided millimeter-wave (mmWave) wireless system and present a covert transmission design from the mmWave-UAV (Alice) to a legitimate ground user (Bob), subject to detection attack by a third party (Willie). Of particular interest is that Alice has no knowledge of both Bob and Willie&#x2019;s exact locations, which are deemed to be difficult to estimate in practice. As a result, Alice opts for a time-division based beam-sweeping transmission scheme to enable opportunistic communication to Bob. An average throughput maximization problem is formulated that jointly optimizes the number of beams, transmit power and flight altitude of Alice, by taking into account the statistical location distribution of Bob and a covertness constraint on Willie. To solve this highly non-concave problem that involves a stochastic objective function, we exploit the minorization-maximization method, transform the problem into a series of simple approximation problems and then develop a batch stochastic minorization-maximization (BSM) algorithm. Numerical results confirm the effectiveness of the BSM algorithm and demonstrate important tradeoff among the key design parameters considered.

Robust motion trajectory optimization of overhead cranes based on polynomial chaos expansion

In this paper, we present a polynomial chaos-based framework for the trajectory optimization of an overhead crane system under uncertainty. The main research described in this paper is as follows. First, the deterministic trajectory optimization problem formulation of a two-dimensional overhead crane model is constructed. Based on this basic mathematical formulation, the uncertainty trajectory optimization problem is formed considering the uncertainty of initial state and system parameter. Then, to solve the uncertainty trajectory optimization problem efficiently, a robust trajectory optimization problem formulation is proposed. However, it is difficult to solve the robust trajectory optimization problem directly because it contains stochastic function terms, such as stochastic dynamic equations, constraint functions and objective functions. We consider both the system state and control input as functions of uncertainty and use polynomial chaos expansion to quantify these stochastic functions. An augmented deterministic trajectory optimization problem which can be solved directly is finally obtained. Based on the proposed robust trajectory optimization formation, the motion trajectory optimization of an overhead crane system under two different uncertainty types of is solved. All simulation results are compared with traditional sampling-based Monte Carlo simulations to demonstrate the feasibility and effectiveness of the proposed method.

MS-NET: modular selective network

We propose a modular architecture of Deep Neural Network (DNN) for multi-class classification task. The architecture consists of two parts, a router network and a set of expert networks. In this architecture, for a C-class classification problem, we have exactly C experts. The backbone network for these experts and the router are built with simple and identical DNN architecture. For each class, the modular network has a certain number rho of expert networks specializing in that particular class, where rho is called the redundancy rate in this study. We demonstrate that rho plays a vital role in the performance of the network. Although these experts are light weight and weak learners alone, together they match the performance of more complex DNNs. We train the network in two phase wherein, first the router is trained on the whole set of training data followed by training each expert network enforced by a new stochastic objective function that facilitates alternative training on a small subset of expert data and the whole set of data. This alternative training provides an additional form of regularization and avoids over-fitting the expert network on subset data. During the testing phase, the router dynamically selects a fixed number of experts for further evaluation of the input datum. The modular nature and low parameter requirement of the network makes it very suitable in distributed and low computational environments. Extensive empirical study and theoretical analysis on CIFAR-10, CIFAR-100 and F-MNIST substantiate the effectiveness and efficiency of our proposed modular network.

Virtual Autonomous Vehicle Using Deep Reinforcement Learning

In this paper, we propose an in-depth reinforcement learning approach to evaluate the performance of the virtually created autonomous vehicle driving scenario. Markov Decision Process is used to map the state of the vehicle to an action. Discount and Reward functions are also incorporated in the decision policy. To deal with high-dimensional inputs which lead to the standard instabilities of reinforcement learning, we have used experience replay. To further reduce the correlation, we use an iterative update that updates the Q-values periodically. Adam Optimizer, based on stochastic objective functions, is used as an optimizer in a neural network along with Rectified Linear Unit activation function, which helps in further optimizing the process. The autonomous vehicle doesn't need any labelled training data to learn human driving behaviour. Inspired by the real-world scenarios, an action-based reward function is used to train the vehicle. It has been shown in our approach that after some iterations, the virtually created vehicle generates collision-free movement and performs the same driving behaviour as of humans.

Optimization Approaches for the Numerical Design of Structures Under Consideration of Polymorphic Uncertain Data

Abstract In this paper, optimization approaches for the numerical design of structures are presented. The uncertainty of structural parameters is considered by means of random variables and interval design parameters within the optimization. A new space subdividing technique is introduced to substitute time-consuming failure probability constraint evaluations. In order to solve optimization problems with polymorphic uncertain parameters, surrogate objectives are formulated and solved by means of a particle swarm optimization (PSO) approach in combination with Monte Carlo simulation and interval analysis. Two computational schemes are presented and verified by a benchmark example. Finally, an application is shown, where the reinforcement layout of a reinforced concrete bridge structure is optimized by minimizing the crack widths at the reinforcement layers in order to improve the durability of the structure. A nonlinear finite element (FE) model is used to compute the uncertain crack patterns and the load bearing capacity. The stochastic objective function and the failure probability constraint are approximated by neural network based surrogate models.