Class Of Optimization Problems Research Articles

ON THE SOLUTION UNIQUENESS IN PORTFOLIO OPTIMIZATION AND RISK ANALYSIS

In this paper, we consider the issue of solution uniqueness of the mean-deviation portfolio optimization problem and its inverse for asset returns distributed over a finite number of scenarios. Due to the asymmetry of returns, the risk is assessed by a general deviation measure introduced by Rockafellar et al. [(2006b) Optimality conditions in portfolio analysis with general deviation measures, Mathematical Programming 108, 515–540] instead of the standard deviation as in the classical Markowitz optimization problem. We demonstrate that, in general, one cannot expect the uniqueness of Pareto-optimal profit sharing in cooperative investment and the uniqueness of solutions in the mean-deviation Black–Litterman asset allocation model. For a large class of deviation measures, we provide a resolution of the above nonuniqueness issues based on the principle of law-invariance. We provide several examples illustrating the nonuniqueness and the law-invariant solution.

Simulating adiabatic quantum computing with parameterized quantum circuits

Abstract Adiabatic quantum computing is a universal model for quantum computing whose implementation using a gate-based quantum computer requires depths that are unreachable in the early fault-tolerant era. To mitigate the limitations of near-term devices, a number of hybrid approaches have been pursued in which a parameterized quantum circuit prepares and measures quantum states and a classical optimization algorithm minimizes an objective function that encompasses the solution to the problem of interest. In this work, we propose a different approach starting by analyzing how a small perturbation of a Hamiltonian affects the parameters that minimize the energy within a family of parameterized quantum states. We derive a set of equations that allow us to compute the new minimum by solving a constrained linear system of equations that is obtained from measuring a series of observables on the unperturbed system. We then propose a discrete version of adiabatic quantum computing that can be implemented in a near-term device while at the same time is insensitive to the initialization of the parameters and to other limitations hindered in the optimization part of variational quantum algorithms. We compare our proposed algorithm with the variational quantum eigensolver on two classical optimization problems, namely MaxCut and number partitioning, and on a quantum-spin configuration problem, the transverse-field ising chain model, and confirm that our approach demonstrates superior performance.

Reinforcement learning-based alpha-list iterated greedy for production scheduling

Metaheuristics can benefit from analyzing patterns and regularities in data to perform more effective searches in the solution space. In line with the emerging trend in the optimization literature, this study introduces the Reinforcement-learning-based Alpha-List Iterated Greedy (RAIG) algorithm to contribute to the advances in machine learning-based optimization, notably for solving combinatorial problems. RAIG uses an N-List mechanism for solution initialization and its solution improvement procedure is enhanced by Reinforcement Learning and an Alpha-List mechanism for more effective searches. A classic engineering optimization problem, the Permutation Flowshop Scheduling Problem (PFSP), is considered for numerical experiments to evaluate RAIG's performance. Highly competitive solutions to the classic scheduling problem are identified, with up to 9% improvement compared to the baseline, when solving large-size instances. Experimental results also show that the RAIG algorithm performs more robustly than the baseline algorithm. Statistical tests confirm that RAIG is superior and hence can be introduced as a strong benchmark for future studies.

Computing optimal drug dosing regarding efficacy and safety: the enhanced OptiDose method in NONMEM

Recently, an optimal dosing algorithm (OptiDose) was developed to compute the optimal drug doses for any pharmacometrics model for a given dosing scenario. In the present work, we enhance the OptiDose concept to compute optimal drug dosing with respect to both efficacy and safety targets. Usually, these are not of equal importance, but one is a top priority, that needs to be satisfied, whereas the other is a secondary target and should be achieved as good as possible without failing the top priority target. Mathematically, this leads to state-constrained optimal control problems. In this paper, we elaborate how to set up such problems and transform them into classical unconstrained optimal control problems which can be solved in NONMEM. Three different optimal dosing tasks illustrate the impact of the proposed enhanced OptiDose method.

Decoding decision delegation to artificial intelligence: A mixed-methods study on the preferences of decision-makers and decision-affected in surrogate decision contexts

Organizations are increasingly opting to use decision-making systems based on artificial intelligence (AI) to increase decision quality and speed and free up resources. However, delegating decisions to AI is challenging, especially in contexts that go beyond classical optimization problems. Current research has so far largely neglected to include the viewpoints of the decision-affected and of surrogate decision-makers in the discussion. To address these gaps, in this study we rely on two experimental designs and complement our findings with qualitative interview data to shed light on both sides: the perspective of those affected by the decision and that of the decision maker in surrogate and ethically complex decision contexts. Findings reveal that the individual decision-making perspective may lead to opposing perceptions. Whereas the willingness to delegate an ethically challenging decision to AI in a surrogate decision context is lower than that in a non-surrogate context (decision-maker perspective), people affected by the decision do not generally prefer humans to AI (decision-affected perspective). By these findings, we contribute to the literature on AI-enabled decision-making and decision delegation to non-human entities.

Feedback-based quantum algorithms for ground state preparation

The ground state properties of quantum many-body systems are a subject of interest across chemistry, materials science, and physics. Thus, algorithms for finding ground states can have broad impacts. Variational quantum algorithms are one class of ground state algorithms that has received significant attention in recent years. These algorithms utilize a hybrid quantum-classical computing framework to prepare ground states on quantum computers. However, this requires solving a classical optimization problem that can become prohibitively expensive in high dimensions. Here, we develop formulations of feedback-based quantum algorithms for ground state preparation that can be used to address this challenge for two broad classes of Hamiltonians: Fermi-Hubbard Hamiltonians, and molecular Hamiltonians represented in second quantization. Feedback-based quantum algorithms are optimization-free; in place of classical optimization, quantum circuit parameters are set according to a deterministic feedback law derived from quantum Lyapunov control principles. This feedback law guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. A variety of numerical illustrations are provided that analyze the convergence and robustness of feedback-based quantum algorithms for these problem classes. Published by the American Physical Society 2024

Graphs with degree sequence [formula omitted] and [formula omitted]

In this paper we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. We establish various properties of Gm,n and G¯m,n related to recognition, connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We also show that several classical optimization problems on these graphs are NP-hard, while others can be solved efficiently.

An accelerated deterministic algorithm for maximizing monotone submodular minus modular function with cardinality constraint

Submodular optimization not only covers some classical combinatorial optimization problems, but also has a wide range of applications in fields such as machine learning and artificial intelligence. For submodular maximization problems with constraints, some work has been done including the design of approximation algorithms, the measurement of approximation algorithms in terms of quality and efficiency, etc. In this paper, we consider the problem of maximizing a non-negative monotone submodular function minus a non-negative modular function with the cardinality constraint. This model has been applied to many scenarios, such as team formation problem, influence maximization problem, recommender systems problem, etc. We propose a threshold algorithm that achieve a (1/2−O(ε),2)-bicriteria approximation ratio and query complexity O(nlog⁡n). Our algorithm makes a small sacrifice in the approximation ratio but improves the best query complexity result of existing deterministic algorithms from O(n2) to O(nlog⁡n) in the worst case.

Optimizers of three-point energies and nearly orthogonal sets

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for p p -frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the p p -frame three-point energy when 0 > p > 1 0>p>1 in all dimensions. The arguments make use of multivariate polynomials employed in semidefinite programming and based on the classical Gegenbauer polynomials. For p = 1 p=1 , we completely solve the analogous problem on the circle. As for higher dimensions, we show that the Hausdorff dimension of minimizers is not greater than d − 2 d-2 for measures on S d − 1 \mathbb {S}^{d-1} .

On symmetry adapted bases in trigonometric optimization

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu Dumitrescu (2007). To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP.The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference Hubert et al. (2022) and later extended to Hubert et al. (2024). In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance.Partial results of this article have been presented as a poster at the ISSAC 2023 conference Metzlaff (2023).

Finding and exploring promising search space for The 0–1 Multidimensional Knapsack Problem

The 0–1, Multidimensional Knapsack Problem (MKP) is a classical NP-hard combinatorial optimization problem with many engineering applications. In this paper, we propose a novel algorithm combining evolutionary computation with the exact algorithm to solve the 0–1 MKP. It maintains a set of solutions and utilizes the information from the population to extract good partial assignments. To find high-quality solutions, an exact algorithm is applied to explore the promising search space specified by the good partial assignments. The new solutions are used to update the population. Thus, the good partial assignments evolve towards a better direction with the improvement of the population. Extensive experimentation with commonly used benchmark sets shows that our algorithm outperforms the state-of-the-art heuristic algorithms, TPTEA and DQPSO, as well as the commercial solver CPlex. It finds better solutions than the existing algorithms and provides new lower bounds for 10 large and hard instances.

Improved Ant Colony Algorithm for the Split Delivery Vehicle Routing Problem

The split delivery vehicle routing problem (SDVRP) is a classic combinatorial optimization problem, which is usually solved using a heuristic algorithm. The ant colony optimization algorithm is an excellent heuristic algorithm that has been successfully applied to solve various practical problems, and it has achieved good results. However, in the existing ant colony optimization algorithms, there are issues with weak targeting of different customer selection strategies, difficulty in balancing convergence speed and global search ability, and a predisposition to become trapped in local optima. To solve these problems, this paper proposes an improved ant colony algorithm (IACA). First, in terms of customer point selection, the initial customer and noninitial customer selection strategies are proposed for different customers, and the adaptive selection threshold is designed. Second, in terms of pheromone processing, an initial pheromone distribution method based on a greedy strategy, a pheromone backtracking mechanism, and an adaptive pheromone volatile factor are proposed. Finally, based on the 2-opt local search method, vehicle path self-search and intervehicle path search are proposed to further improve the quality of the solution. This paper tests the performance of the IACA on datasets of different scales. The experimental results show that compared with the clustering algorithm, artificial bee colony algorithm, particle swarm optimization algorithm, traditional ant colony algorithm, and other algorithms, the IACA can achieve more competitive results. Specifically, compared to the path length calculated by other algorithms, the path length calculated by IACA decreased by an average of 1.58%, 4.28%, and 3.64% in small, medium, and large-scale tests, respectively.

A Metaheuristic Framework with Experience Reuse for Dynamic Multi-Objective Big Data Optimization

Dynamic multi-objective big data optimization problems (DMBDOPs) are challenging because of the difficulty of dealing with large-scale decision variables and continuous problem changes. In contrast to classical multi-objective optimization problems, DMBDOPs are still not intensively explored by researchers in the optimization field. At the same time, there is lacking a software framework to provide algorithmic examples to solve DMBDOPs and categorize benchmarks for relevant studies. This paper presents a metaheuristic software framework for DMBDOPs to remedy these issues. The proposed framework has a lightweight architecture and a decoupled design between modules, ensuring that the framework is easy to use and has enough flexibility to be extended and modified. Specifically, the framework now integrates four basic dynamic metaheuristic algorithms, eight test suites of different types of optimization problems, as well as some performance indicators and data visualization tools. In addition, we have proposed an experience reuse method, speeding up the algorithm’s convergence. Moreover, we have implemented parallel computing with Apache Spark to enhance computing efficiency. In the experiments, algorithms integrated into the framework are tested on the test suites for DMBDOPs on an Apache Hadoop cluster with three nodes. The experience reuse method is compared to two restart strategies for dynamic metaheuristics.

Optimal solving of a binary knapsack problem on a D-Wave quantum machine and its implementation in production systems

AbstractThe efficient management of complex production systems is a challenge in today’s logistics. In the field of intelligent and sustainable logistics, the optimization of production batches, especially in the context of a rapidly changing product range, requires fast and precise computational solutions. This paper explores the potential of quantum computers for solving these problems. Traditional computational methods are often limited when it comes to optimizing complex logistics systems. In response to these challenges, the paper proposes the use of a hybrid algorithm that combines quantum technologies with classical computational methods. Such integration allows the computational power of both types of technologies to be harnessed, leading to faster and more efficient identification of optimal solutions. In this work, we consider the knapsack problem, a classic NP-hard optimization problem that is commonly used to verify the effectiveness of new algorithm construction methods. The algorithm presented is based on the Branch and Bound method and aims to ensure solution optimality in the context of the non-determinism of quantum computers. Within the algorithm, computations are performed alternately on a classical processor and a quantum processor. In addition, the lower and upper bounds of the objective function are computed in constant time using the D-Wave quantum machine.

On conflict-free cuts: Algorithms and complexity

One way to define the Matching Cut problem is: Given a graph G, is there an edge-cut M of G such that M is an independent set in the line graph of G? We propose the more general Conflict-Free Cut problem: Together with the graph G, we are given a so-called conflict graph Gˆ on the edges of G, and we ask for an edge-cutset M of G that is independent in Gˆ. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is NP-complete even when the maximum degree of G is 5 and Gˆ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of G as a parameter, and we show W[1]-hardness even when G has a feedback vertex set of size one, and the clique cover number of Gˆ is the parameter. Since the clique cover number of Gˆ is an upper bound on the independence number of Gˆ and thus the solution size, this implies W[1]-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT.

Кількісне оцінювання функціональної ефективності мережевих структур за аналізом чутливості оптимізаційної моделі до зовнішніх впливів

The real network organizational structure is constantly under the influence of external influences, which can cause unexpected consequences due to the cascading spread of relevant disturbances directed at critically important parameters. Parametric sensitivity analysis is the latest computational procedure, the results of which provide the researcher with an idea and quantitative assessment of the appropriate response of the structure in the form of an analytical platform as a composition and interaction of scenario analysis, cascade process development and optimization modeling technology. The main task is timely adaptation to expected changes in business analytics, oriented to optimal resources distribution. The generalized minimum cost problem is a typical example of widespread problems of optimal distribution and use of limited, in particular, energy resources, of any nature and purpose, using the scientific management and analysis of decisions, in its network model attention is paid to the potentials of nodes. The maximum flow problem is an example of the active and effective application of methods and models of flows optimization, where the processes and objects used a network organization, the energy systems and complexes investigated, where arc parameters are critical in its model. In the article, for these classic optimization problems, mathematical and spreadsheet models were built on concrete examples. Direct and dual mathematical programming problems were solved, the following stages were carried out - the organization of a computer experiment and the construction of an analytical platform for evaluating the behavior of the network structure under the influence of external influences and disturbances as their consequences. To take into account the specific conditions of real objects with a network structure at the stage of modification, the model supplemented with appropriate restrictions and correction of input data sets. The obtained results should be useful for planning and management personnel and top-management for discussion and decision-making regarding, for example, the optimal placement of energy equipment, the weighted distribution of energy flows in the "source-sink" system or the appointment of executors for work, automated project management using project networks, etc. Keywords: optimization modeling, maximum network flow, minimum cost flows, network organizational structures, decision making, spreadsheet modeling and analytics.

Two-Stage Probe-Based Search Optimization Algorithm for the Traveling Salesman Problems

As a classical combinatorial optimization problem, the traveling salesman problem (TSP) has been extensively investigated in the fields of Artificial Intelligence and Operations Research. Due to being NP-complete, it is still rather challenging to solve both effectively and efficiently. Because of its high theoretical significance and wide practical applications, great effort has been undertaken to solve it from the point of view of intelligent search. In this paper, we propose a two-stage probe-based search optimization algorithm for solving both symmetric and asymmetric TSPs through the stages of route development and a self-escape mechanism. Specifically, in the first stage, a reasonable proportion threshold filter of potential basis probes or partial routes is set up at each step during the complete route development process. In this way, the poor basis probes with longer routes are filtered out automatically. Moreover, four local augmentation operators are further employed to improve these potential basis probes at each step. In the second stage, a self-escape mechanism or operation is further implemented on the obtained complete routes to prevent the probe-based search from being trapped in a locally optimal solution. The experimental results on a collection of benchmark TSP datasets demonstrate that our proposed algorithm is more effective than other state-of-the-art optimization algorithms. In fact, it achieves the best-known TSP benchmark solutions in many datasets, while, in certain cases, it even generates solutions that are better than the best-known TSP benchmark solutions.

Circumventing superexponential runtimes for hard instances of quantum adiabatic optimization

Classical optimization problems can be solved by adiabatically preparing the ground state of a quantum Hamiltonian that encodes the problem. The performance of this approach is determined by the smallest gap encountered during the evolution. Here, we consider the maximum independent set problem, which can be efficiently encoded in the Hamiltonian describing a Rydberg atom array. We present a general construction of instances of the problem for which the minimum gap decays superexponentially with system size, implying a superexponentially large time to solution via adiabatic evolution. The small gap arises from locally independent choices which cause the system to initially evolve and localize into a configuration far from the solution in terms of Hamming distance. We investigate remedies to this problem. Specifically, we show that quantum quenches in these models can exhibit signatures of quantum many-body scars, which in turn, can circumvent the superexponential gaps. By quenching from a suboptimal configuration, states with a larger ground-state overlap can be prepared, illustrating the utility of quantum quenches as an algorithmic tool. Published by the American Physical Society 2024

Quantum approximate optimization via learning-based adaptive optimization

Combinatorial optimization problems are ubiquitous and computationally hard to solve in general. Quantum approximate optimization algorithm (QAOA), one of the most representative quantum-classical hybrid algorithms, is designed to solve combinatorial optimization problems by transforming the discrete optimization problem into a classical optimization problem over continuous circuit parameters. QAOA objective landscape is notorious for pervasive local minima, and its viability significantly relies on the efficacy of the classical optimizer. In this work, we design double adaptive-region Bayesian optimization (DARBO) for QAOA. Our numerical results demonstrate that the algorithm greatly outperforms conventional optimizers in terms of speed, accuracy, and stability. We also address the issues of measurement efficiency and the suppression of quantum noise by conducting the full optimization loop on a superconducting quantum processor as a proof of concept. This work helps to unlock the full power of QAOA and paves the way toward achieving quantum advantage in practical classical tasks.

A branch, bound and remember algorithm for maximizing the production rate in the simple assembly line balancing problem

The simple assembly line balancing problem is a classical optimization problem. It aims at efficiently assigning elementary operations to the workstations of a paced assembly line to optimize some efficiency measure. In this paper, we deal with the type-2 version of the problem in which the efficiency measure is the maximization of throughput with a given number of workstations. This problem has been scarcely studied compared to its type-1 counterpart, in which the throughput is fixed and the objective is to minimize the number of workstations. This has led to the development of indirect methods using type-1 exact solution methods to tackle the type-2 case.In this paper, we provide lower bounds, logical tests, and dominance rules for the type-2 problem and adapt the state-of-the-art method for the type-1 problem to develop a direct approach for solving the type-2 problem with a direct approach. Computational results show the effectiveness of the direct method, which can outperform recent type-1-based indirect approaches. The proposed method is able to close all instances from the classical instance set and solves to optimality over 90% of the large (100 tasks) and 76.5% of the very large (1000 tasks) instances derived from the current reference set for its type-1 counterpart. The average gaps between the upper and lower bounds on the optimal cycle time are also tight and are less than 0.13% (0.4%) for large (very large) instances. Finally, our results show that direct and indirect methods are comparable in terms of solution quality, but the direct method outperforms its indirect counterpart in terms of running time and optimality gaps. Moreover, we identify that instances averaging two to three tasks per workstation and featuring large variability between task times pose significant challenges to solve to optimality.