General Optimization Problem Research Articles

Single machine robust scheduling with budgeted uncertainty

We consider robust single machine scheduling problems. First, we prove that with uncertain processing times, minimizing the number of tardy jobs is NP-hard. Second, we show that the weighted variant of the problem has the same complexity as the nominal counterpart whenever only the weights are uncertain. Last, we provide approximation algorithms for the problems minimizing the weighted sum of completion times. Noticeably, our algorithms extend to more general robust combinatorial optimization problems with cost uncertainty, such as max-cut.

Multiple sparse detection-based evolutionary algorithm for large-scale sparse multiobjective optimization problems

AbstractSparse multiobjective optimization problems are common in practical applications. Such problems are characterized by large-scale decision variables and sparse optimal solutions. General large-scale multiobjective optimization problems (LSMOPs) have been extensively studied for many years. They can be well solved by many excellent custom algorithms. However, when these algorithms are used to deal with sparse LSMOPs, they often encounter difficulties because the sparse nature of the problem is not considered. Therefore, aiming at sparse LSMOPs, an algorithm based on multiple sparse detection is proposed in this paper. The algorithm applies an adaptive sparse genetic operator that can generate sparse solutions by detecting the sparsity of individuals. To improve the deficiency of sparse detection caused by local detection, an enhanced sparse detection (ESD) strategy is proposed in this paper. The strategy uses binary coefficient vectors to integrate the masks of nondominated solutions. Essentially, the mask is globally and deeply optimized by coefficient vectors to enhance the sparsity of the solutions. In addition, the algorithm adopts an improved weighted optimization strategy to fully optimize the key nonzero variables to balance exploration and optimization. Finally, the proposed algorithm is named MOEA-ESD and is compared to the current state-of-the-art algorithm to verify its effectiveness.

Resetting-mediated navigation of an active Brownian searcher in a homogeneous topography.

Designing navigation strategies for search-time optimization remains of interest in various interdisciplinary branches in science. Herein, we focus on active Brownian walkers in noisy and confined environments, which are mediated by one such autonomous strategy, namely stochastic resetting. As such, resetting stops the motion and compels the walkers to restart from the initial configuration intermittently. The resetting clock is operated externally without any influence from the searchers. In particular, the resetting coordinates are either quenched (fixed) or annealed (fluctuating) over the entire topography. Although the strategy relies upon simple governing laws of motion, it shows a significant ramification for the search-time statistics, in contrast to the search process conducted by the underlying reset-free dynamics. Using extensive numerical simulations, we show that the resetting-driven protocols enhance the performance of these active searchers. This, however, depends robustly on the inherent search-time fluctuations, measured by the coefficient of variation of the underlying reset-free process. We also explore the effects of different boundaries and rotational diffusion constants on the search-time fluctuations in the presence of resetting. Notably, for the annealed condition, resetting is always found to expedite the search process. These features, as well as their applicability to more general optimization problems from queuing systems, computer science and randomized numerical algorithms, to active living systems such as enzyme turnover and backtracking recovery of RNA polymerases in gene expression, make resetting-based strategies universally promising.

Physics-Constrained Robustness Evaluation of Intelligent Security Assessment for Power Systems

Machine learning (ML) algorithms have been widely developed to enable real-time security assessment for large-scale electricity grids. However, it also has been extensively recognized that the ML models are vulnerable to adversarial examples, which are slightly perturbed instances that can mislead the classifications but cannot be distinguished by human eyes. Considering the adversarial examples against the ML-based security assessment, they must not only cause misclassification but also follow the power balance and transfer limits and bypass the bad data detection. To this end, this paper proposes a novel concept named physics-constrained robustness that aims to computing a lower-bound of adversarial perturbations, evaluating the vulnerability of the ML-based intelligent security assessment (ISA) for power systems. A general optimization problem is formulated to compute the physics-constrained robustness of ISA with the mislabeling, power balance, power limitation, and invisible constraints. The analytical results and comparing relationship of ISA’s robustness with different constraints are provided. By using the static security assessment as an example, we provide explicit formulations to evaluate the physics-constrained robustness of. Finally, extensive experiments are conducted to evaluate the physics-constrained robustness of ISA in static and dynamic cases with the real-world load profiles from New York State and provide suggestions to select the ML models and parameters.

Variations of $v$-change of time in an optimal control problem with state and mixed constraints

For a general optimal control problem with state and regular mixed constraints we propose a proof of the maximum principle based on the so-called $v$-change of time variable $t \mapsto \tau$, under which the original time becomes an additional state variable subject to the equation $dt/d\tau = v(\tau)$, while the additional control variable $v(\tau)\geqslant 0$ is piecewise constant, and its values become arguments of the new problem.

Sequential Convex Programming Methods for Real-Time Optimal Trajectory Planning in Autonomous Vehicle Racing

Optimization problems for trajectory planning in autonomous vehicle racing are characterized by their nonlinearity and nonconvexity. Instead of solving these optimization problems, usually a convex approximation is solved instead to achieve a high update rate. We present a real-time-capable model predictive control (MPC) trajectory planner based on a nonlinear single-track vehicle model and Pacejka’s magic tire formula for autonomous vehicle racing. After formulating the general nonconvex trajectory optimization problem, we form a convex approximation using sequential convex programming (SCP). The state of the art convexifies track constraints using sequential linearization (SL), which is a method of relaxing the constraints. Solutions to the relaxed optimization problem are not guaranteed to be feasible in the nonconvex optimization problem. We propose sequential convex restriction (SCR) as a method to convexify track constraints. SCR guarantees that resulting solutions are feasible in the nonconvex optimization problem. We show recursive feasibility of solutions to the restricted optimization problem. The MPC is evaluated on a scaled version of the Hockenheimring racing track in simulation. The results show that MPC using SCR yields faster lap times than MPC using SL, while still being real-time capable.

Asymptotic analysis of scalarization functions and applications

<p style='text-indent:20px;'>In this paper, we consider two common scalarization functions and their applications via asymptotic analysis. We mainly analyze the recession and asymptotic properties of translation invariant function and oriented distance function, and discuss their monotonicity and Lipschitz continuity in terms of recession functions. Finally, we apply these scalarization functions to the characterization of the nonemptiness and boundedness of the solution set for a general constrained nonconvex optimization problem.</p>

Optimal control problems with time inconsistency

<abstract><p>In the present study, the necessary and sufficient conditions of equilibrium control for general optimal control problems with time inconsistency are established in sense of open-loop. As an application, the linear quadratic optimal control problems with time inconsistency were also explored and an explicit equilibrium control is constructed.</p></abstract>

A Non-dimensional Input Excitation Optimization Approach for Battery Health Parameter Estimation

Model parameter estimation is an important subject in control engineering, including the field of battery management. Input excitation optimization has become an emerging topic lately to improve the accuracy of estimation. Traditional optimization approach suffers from a fundamental issue in parameter uncertainty, as the target parameters for estimation, often needed for computing the optimization objective and constraints, are intrinsically unknown. In this study, we introduce a non-dimensional approach to optimize excitations for estimating the health-related Li-ion battery electrochemical parameters. Guided by the Buckingham π theorem, we derived a control-oriented non-dimensional battery model, excluding uncertain target parameters from the problem formulation. As a result, the optimization problem can be solved without any prior knowledge of target parameters. The applicable control input sequence can be recovered by rescaling the obtained non-dimensional sequence with the best available knowledge of the parameters. Furthermore, the proposed method reveals the fundamental impact of the unknown parameters on the solution of input optimization. In light of this finding, we propose two iterative excitation optimization strategies, which both significantly improve the robustness and reduce the complexity of the optimization problem. The proposed method can be generalized to solve general optimal control problems for a broad class of systems.

Integration of Simulated Quantum Annealing in Parallel Tempering and Population Annealing for Heterogeneous-Profile QUBO Exploration

Simulated Quantum Annealing (SQA) is a heuristic algorithm which can solve Quadratic Unconstrained Binary Optimization (QUBO) problems by emulating the exploration of the solution space done by a quantum annealer. It mimics the quantum superposition and tunnelling effects through a set of correlated replicas of the spins system representing the problem to be solved and performing Monte Carlo steps. However, the effectiveness of SQA over a classical algorithm strictly depends on the cost/energy profile of the target problem. In fact, quantum annealing only performs well in exploring functions with high and narrow peaks, while classical annealing is better in overcoming flat and wide energy-profile barriers. Unfortunately, real-world problems have a heterogeneous solution space and the probability of success of each solver depends on the size of the energy profile region compatible with its exploration mechanism. Therefore, significant advantages could be obtained by exploiting hybrid solvers, which combine SQA and classical algorithms. This work proposes four new quantum-classical algorithms: Simulated Quantum Parallel Tempering (SQPT), Simulated Quantum Population Annealing (SQPA), Simulated Quantum Parallel Tempering - Population Annealing v1 (SQPTPA1) and Simulated Quantum Parallel Tempering - Population Annealing v2 (SQPTPA2). They are obtained by combining SQA, Parallel Tempering (PT), and Population Annealing (PA). Their results are compared with those provided by SQA, considering benchmark QUBO problems, characterized by different profiles. Even though this work is preliminary, the obtained results are encouraging and prove hybrid solvers’ potential in solving a generic optimization problem.

A Novel Approach to Integrating Uncertainty into a Push Re-Label Network Flow Algorithm for Pit Optimization

The standard optimization of open-pit mine design and production scheduling, which is impacted by a variety of factors, is an essential part of mining activities. The metal uncertainty, which is connected to supply uncertainty, is a crucial component in optimization. To address uncertainties regarding the economic value of mining blocks and the general problem of mine design optimization, a minimum-cut network flow algorithm is employed to give the optimal ultimate pit limits and pushback designs under uncertainty. A structure that is computationally effective and can manage the joint presentation and treatment of the economic values of mining blocks under various circumstances is created by the push re-label minimum-cut technique. In this study, the algorithm is put to the test using a copper deposit and shows similarities to other stochastic optimizers for mine planning that have already been created. Higher possibilities of reaching predicted production targets are created by the algorithm’s earlier selection of more certain blocks with blocks of high value. Results show that, in comparison to a conventional approach using the same algorithm, the cumulative metal output is larger when the uncertainty in the metal content is taken into consideration. There is also an additional 10% gain in net present value.

End-to-End Service Auction: A General Double Auction Mechanism for Edge Computing Services

Ubiquitous powerful personal computing facilities, such as desktop computers and parked autonomous cars, can function as micro edge computing servers by leveraging their spare resources. However, to harvest their resources for service provisioning, two significant challenges will arise: how to incentivize the server owners to contribute their computing resources, and how to guarantee the end-to-end (E2E) Quality-of-Service (QoS) for service buyers? In this paper, we address these two problems in a holistic way by advocating COMSA. Unlike the existing double auction schemes for edge computing which mostly focus on computing resource trading, COMSA addresses the joint problem of double auction mechanism design and network resource allocation by explicitly taking spectrum allocation and data routing into account, thereby providing E2E QoS guarantees for edge computing services. To handle the design complexity, COMSA employs a two-step procedure to decouple network optimization and mechanism design, which hence can be applied to general network optimization problems for edge computing. COMSA holds some critical economic properties, i.e., truthfulness, budget balance, and individual rationality. Our extensive simulation studies demonstrate the effectiveness of COMSA.

A surrogate-assisted variable grouping algorithm for general large-scale global optimization problems

Problem decomposition plays an important role when applying cooperative coevolution (CC) to large-scale global optimization problems. However, most learning-based decomposition algorithms only apply to additively separable problems, while the others insensitive to problem type perform low decomposition accuracy and efficiency. Given this limitation, this study designs a general-separability-oriented detection criterion, and further proposes a novel decomposition algorithm called surrogate-assisted variable grouping (SVG). The new criterion detects the separability between a variable and some other variables by checking whether its optimum changes with the latter. Consistent with the definition of general separability, this criterion endows SVG with strong applicability and high accuracy. To reduce expensive fitness evaluations, SVG locates the optimum of a variable with the help of a surrogate model rather than the original high-dimensional model. Moreover, it converts the variable-grouping process into a search process in a binary tree by taking variable subsets as tree nodes. This facilitates the reutilization of historical separability information, thereby reducing separability detection times. Experimental results on a general benchmark suite indicate that compared with six state-of-the-art decomposition algorithms, SVG achieves higher accuracy and efficiency on both additively and nonadditively separable problems. Furthermore, it can significantly enhance the optimization performance of CC.

Generating Nesterov’s accelerated gradient algorithm by using optimal control theory for optimization

Nesterov’s accelerated gradient algorithm is derived from first principles. The first principles are founded on the recently-developed optimal control theory for optimization. This theory frames a generic optimization problem as an optimal control problem. The necessary conditions for optimal control produce a controllable dynamical system. Stabilizing this system by dissipating a simple, quadratic “control” Lyapunov function generates an equation that produces Nesterov’s algorithm. In this context, the proposed theory solves the long-standing mystery surrounding the algorithm.

Solving combinatorial optimization problems over graphs with BERT-Based Deep Reinforcement Learning

Combinatorial optimization, such as vehicle routing and traveling salesman problems for graphs, is NP-hard and has been studied for decades. Many methods have been proposed for its possible solution, including, but not limited to, exact algorithms, approximate algorithms, heuristic algorithms, and solution solvers. However, these methods cannot learn the problem’s internal structure nor generalize to similar or larger-scale problems. Recently, deep reinforcement learning has been applied to combinatorial optimization and has achieved convincing results. Nevertheless, the challenge of effective integration and training improvement still exists. In this study, we propose a novel framework (BDRL) that combines BERT (Bidirectional Encoder Representations from Transformers) and deep reinforcement learning to tackle combinatorial optimization over graphs by treating general optimization problems as data points under an identified data distribution. We first improved the transformer encoder of BERT to embed the combinatorial optimization graph effectively. By employing contrastive objectives, we extend BERT-like training to reinforcement learning and acquire self-attention-consistent representations. Next, we used hierarchical reinforcement learning to pre-train our model; that is, to train and fine-tune the model through an iterative process to make it more suitable for a specific combinatorial optimization problem. The results demonstrate our proposed framework’s generalization ability, efficiency, and effectiveness in multiple tasks.

Multi-objective low-thrust spacecraft trajectory design using reachability analysis

One of the fundamental problems in spacecraft trajectory design is finding the optimal transfer trajectory that minimizes the propellant consumption and transfer time simultaneously. We formulate this as a multi-objective optimal control (MOC) problem that involves optimizing over the initial or final state, subject to state constraints. Drawing on recent developments in reachability analysis subject to state constraints, we show that the proposed MOC problem can be stated as an optimization problem subject to a constraint that involves the sub-level set of the viscosity solution of a quasi-variational inequality. We then generalize this approach to account for more general optimal control problems in Bolza form. We relate these problems to the Pareto front of the developed multi-objective programs. The proposed approach is demonstrated on two low-thrust orbital transfer problems around a rotating asteroid.

Duality of optimization problems with gauge functions

ABSTRACT Recently, Yamanaka and Yamashita proposed the so-called positively homogeneous optimization problem, which includes many important problems, such as the absolute-value and the gauge optimization problems. They presented a closed form of the dual formulation for the problem, and showed weak duality and the equivalence to the Lagrangian dual under some conditions. In this work, we focus on a special positively homogeneous optimization problem, whose objective function and constraints consist of some gauge and linear functions. We prove not only weak duality but also strong duality. We also study necessary and sufficient optimality conditions associated to the problem. Moreover, we give sufficient conditions under which we can recover a primal solution from a Karush-Kuhn-Tucker point of the dual formulation. Finally, we discuss how to extend the above results to general optimization problems by considering the so-called perspective functions.

From Perspective Maps to Epigraphical Projections

The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus, the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map—including local Lipschitz continuity, directional differentiability, and semismoothness—are derived. An SC1 optimization framework for computing epigraphical and level-set projections is, thus, established. Numerical experiments on one-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC). M.P. Friedlander was supported by NSERC discovery grant [Grant RGPIN-2017-04461]. T. Hoheisel was supported by NSERC discovery grant [Grant RGPIN-2017-04035]. A. Goodwin’s work was partially supported by an NSERC summer research stipend.

An algorithm based on an iterative optimal stopping method for Feller processes with applications to impulse control, perturbation, and possibly zero random discount problems

In this paper we present an iterative optimal stopping method for general optimal stopping problems for Feller processes. We show using an approximating scheme that the value function of an optimal stopping problem for some general operator is the unique viscosity solution to an Hamilton–Jacobi–Bellman equation (see for example Theorems 2.3 and 2.4). We apply our results to study impulse control problems for Feller–Markov processes and derive explicit solutions in the case of one dimensional regular Feller diffusion. We also use our result to study optimal stopping problems for both regime switching and semi-Markov processes and characterize their value functions as the limit of iterative optimal stopping problems (see Corollary 4.2 and Proposition 4.3 ). Finally, we examine optimal stopping problems for random (possibly zero) discount.

Solving Bilevel Optimization Problems Using Kriging Approximations.

Bilevel optimization involves two levels of optimization, where one optimization problem is nested within the other. The structure of the problem often requires solving a large number of inner optimization problems that make these kinds of optimization problems expensive to solve. The reaction set mapping and the lower level optimal value function mapping are often used to reduce bilevel optimization problems to a single level; however, the mappings are not known a priori, and the need is to be estimated. Though there exist a few studies that rely on the estimation of these mappings, they are often applied to problems where one of these mappings has a known form, that is, piecewise linear, convex, etc. In this article, we utilize both these mappings together to solve general bilevel optimization problems without any assumptions on the structure of these mappings. Kriging approximations are created during the generations of an evolutionary algorithm, where the population members serve as the samples for creating the approximations. One of the important features of the proposed algorithm is the creation of an auxiliary optimization problem using the Kriging-based metamodel of the lower level optimal value function that solves an approximate relaxation of the bilevel optimization problem. The auxiliary problem when used for local search is able to accelerate the evolutionary algorithm toward the bilevel optimal solution. We perform experiments on two sets of test problems and a problem from the domain of control theory. Our experiments suggest that the approach is quite promising and can lead to substantial savings when solving bilevel optimization problems. The approach is able to outperform state-of-the-art methods that are available for solving bilevel problems, in particular, the savings in function evaluations for the lower level problem are substantial with the proposed approach.