Proof Of Optimality Research Articles

A Modal Type Theory of Expected Cost in Higher-Order Probabilistic Programs

The design of online learning algorithms typically aims to optimise the incurred loss or cost , e.g., the number of classification mistakes made by the algorithm. The goal of this paper is to build a type-theoretic framework to prove that a certain algorithm achieves its stated bound on the cost. Online learning algorithms often rely on randomness, their loss functions are often defined as expectations, precise bounds are often non-polynomial (e.g., logarithmic) and proofs of optimality often rely on potential-based arguments. Accordingly, we present pλ-amor, a type-theoretic graded modal framework for analysing (expected) costs of higher-order probabilistic programs with recursion. pλ-amor is an effect-based framework which uses graded modal types to represent potentials, cost and probability at the type level. It extends prior work (λ-amor) on cost analysis for deterministic programs. We prove pλ-amor sound relative to a Kripke step-indexed model which relates potentials with probabilistic coupling. We use pλ-amor to prove cost bounds of several examples from the online machine learning literature. Finally, we describe an extension of pλ-amor with a graded comonad and describe the relationship between the different modalities.

An exact constraint programming method for the multi-manned assembly line balancing problem with assignment restrictions

This paper proposes an exact Constraint Programming (CP) method with an extensive focus on real-world constraints for the Multi-manned Assembly Line Balancing Problem with Assignment Restrictions (MALBP-AR). We perform an in-depth literature review to gather examples from real assembly lines and organize the AR regarding tasks, stations, workers, and mounting positions. Our study classifies the AR related to transformed resources and provides a general and unified model. We explore the concept of variable workplaces to dynamically assign workers to mounting positions and aggregate no overlap restrictions to avoid interference between workers. The classic MALBP model is extended by gradually incrementing the number of restrictions. The model variant found in the literature is herein called Partial MALBP-AR. Compared to the previous state-of-the-art Tabu Search Algorithm (TSA) for this problem, the Partial MALBP-AR found twelve additional optimality proofs. Besides the relevant results regarding solution quality, the CP method also has a satisfactory CPU performance. We also propose an entirely new set of AR and test these practical conditions with the so-called Extended MALBP-AR. Such an extended model, which covers all the AR presented here, reached optimality within the computational time limit for 36 out of 38 instances. The worst-case gap for an open instance is 8.20%. The results show a trade-off between the number of deemed restrictions and the computational performance. However, considering the detailed set of AR, we can obtain more representative solutions regarding the final balancing implementation compared to theoretical cases. The method can be used to design experiments, turning certain constraints on and off and allowing managers to evaluate different resource allocation scenarios.

Sorting Permutations on an n − Broom

With applications in computer networks, robotics, genetics, data center network optimization, cryptocurrency exchange, transportation and logistics, cloud computing, and social network analysis, the problem of sorting permutations on transposition trees under various operations is highly relevant. The goal of the problem is to sort or rearrange the markers in a predetermined order by swapping them out at the vertices of a tree in the fewest possible swaps. Only certain classes of transposition trees, like path, star, and broom, have computationally efficient algorithms for sorting permutations. In this paper, we examine the so-called n−broom transposition trees. A single broom or simply a broom is a spanning tree formed by joining the center of the star graph with one end of the path graph. A generalized version of a broom known as an n−broom is created by joining the ends of n brooms to one vertex, known as the n−broom center. By using the idea of clear path markers, we present a novel algorithm for sorting permutations on an n−broom for n>2 that reduces to a novel 2−broom algorithm and that further reduces to two instances of a 1−broom algorithm. Our single-broom algorithm is similar to that of Kawahara et al.; however, our proof of optimality for the same is simpler.

A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization

Wasserstein distributionally robust optimization has emerged as a recent topic with broader applications in operations research and machine learning. Various proofs have been presented in the literature, each differing in assumptions and levels of generality. In “A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization,” Zhang, Yang, and Gao present a novel elementary proof that not only shortens existing frameworks but also offers surprising generalizations. Leveraging classical Legendre—Fenchel duality, they demonstrate that strong duality is contingent on a certain interchangeability principle. Moreover, they extend this duality result to encompass risk-averse optimization and globalized distributionally robust counterparts.

A proven optimal result for a benchmark instance of the uncapacitated examination timetabling problem

AbstractExamination timetabling is a problem well known to the scheduling community. Its simplest version, which is the uncapacitated examination timetabling problem, is easily described and comprehended. Nevertheless, proof of optimality is notoriously difficult even for moderate size problems. In this paper, we describe the effort that our team exercised in finally proving the optimality of the sta83 instance of Carter’s dataset. The problem was decomposed naturally in three parts and for each part a different approach managed to prove optimality of the currently best known solution. This work also presents optimal solutions to subproblems that exist in various public datasets problems and two best known solutions of such problems.

Distributed Resilient Initialization-Free Jacobi Descent Algorithm for Constrained Optimization Against DoS Attacks

This paper investigates one type of distributed constrained optimization problem, e.g., the economic dispatch problem, in the presence of DoS attacks. Therein, multiple DoS attackers are collaborative to impede the communication transmission and change the communication topology at will. Consequently, the convergence and/or optimality of distributed algorithm may be compromised. To reduce the effect of this kind of DoS attacks, a distributed resilient initialization-free Jacobi descent algorithm is proposed. It is designed with three switched control protocols which enable the proposed algorithm reasonably employing the estimations to replace the missing information when attacks occur. Meanwhile, the proposed method is embedded with second order information, resulting in faster convergence speed. Moreover, theoretical analysis results are provided to show that the proposed algorithm can exponentially converge to the global optimal solution of the studied problem. Finally, simulation results tested in IEEE 30-bus system validate its effectiveness and flexibility. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The economic dispatch is a key issue in smart grid, which can be formulated as a kind of distributed constrained optimization problem. Since the distributed algorithms work under distributed sensor networks, they are easier to undergo DoS attacks. To address this issue, this paper presents a distributed resilient initialization-free Jacobi descent algorithm, which features strong robustness to resist DoS attacks and faster convergence. Meanwhile, the proposed method is shaped for common constrained optimization problem with better expansibility. We conduct the global convergence and optimality proofs, which benefits the practitioners to estimate the convergence performance, e.g., the convergence rate. Simulations further show the correctness and effectiveness of the proposed method. In future, we will pay more attention on the non-convex constrained optimization problem.

Optimal teleportation fidelity and its deviation in noisy scenarios

In this work, we study quantum teleportation, a fundamental protocol of quantum physics. In particular, we present a mathematical methodology to study the combined effects of noisy resource state and noisy classical communication on teleportation fidelity and its deviation. We describe a teleportation protocol where an arbitrary two-qubit state in canonical form is used as a resource. Thereafter, to teleport an unknown qubit, Alice measures her qubits in Bell basis and conveys the measurement outcome to Bob via noisy classical channel(s). We derive the exact formulae of optimal teleportation fidelity and corresponding fidelity deviation where the resource state and the classical communication, both can be noisy. To provide the proof of optimality, we provide a systematic method. We further find conditions for non-classical fidelity and dispersion-free teleportation for our teleportation protocol. In this way, we identify noisy environments where it is possible to achieve dispersion-free teleportation without compromising non-classical fidelity. We also demonstrate scenarios where the increase of entanglement in the resource state, may degrade the quality of teleportation — a counter-intuitive instance. Finally, we discuss about the minimum classical communication cost required to achieve non-classical fidelity for our protocol. Here we mainly focus on a possible way to optimize the classical communication cost.

An exact constraint programming based procedure for the multi-manned assembly line balancing problem

Unlike the Simple Assembly Line Balancing Problem (SALBP), the Multi-manned Assembly Line Balancing Problem (MALBP) allows the assignment of multiple workers at the same station. This study proposes a novel Constraint Programming (CP) model to evaluate the satisfiability of a type-F MALBP. Given a cycle time limit, the number of workers and the number of stations are set as parameters in each iteration. We adopt an exact lexicographical procedure as the solution method for this multi-objective optimization problem, performing a lower-bound search to minimize the number of workers as the primary objective. Such an assumption is common for real-world applications, in which the total cost of a worker is usually several orders of magnitude higher than the cost of a station. When dealing with hard instances, we adopt the task–worker assignment solution for SALBP as input to an auxiliary Mixed-Integer Linear Programming (MILP) model, which provides an initial condition for the task scheduling portion of the CP model. Strong tasks’ scheduling bounds are also provided to tighten the search space, and mathematical similarities with SALBP are exploited to support our solution method. The integrated MILP-CP procedure yielded 126 optimal solutions from a 140-instance dataset, including nine new optimality proofs. Moreover, twelve additional instances were improved compared to the literature, enhancing the solution quality. Considering the best-found primal bounds, the maximum gap for the still-open instances is just 0.32%. These results demonstrated the effectiveness of the proposed procedure in solving MALBP when combined with solid bounds.

Asymptotically Optimal Adversarial Strategies for the Probability Estimation Framework.

The probability estimation framework involves direct estimation of the probability of occurrences of outcomes conditioned on measurement settings and side information. It is a powerful tool for certifying randomness in quantum nonlocality experiments. In this paper, we present a self-contained proof of the asymptotic optimality of the method. Our approach refines earlier results to allow a better characterisation of optimal adversarial attacks on the protocol. We apply these results to the (2,2,2) Bell scenario, obtaining an analytic characterisation of the optimal adversarial attacks bound by no-signalling principles, while also demonstrating the asymptotic robustness of the PEF method to deviations from expected experimental behaviour. We also study extensions of the analysis to quantum-limited adversaries in the (2,2,2) Bell scenario and no-signalling adversaries in higher (n,m,k) Bell scenarios.

Optimal consensus control for multi‐agent systems: Multi‐step policy gradient adaptive dynamic programming method

AbstractThis paper presents a novel adaptive dynamic programming (ADP) method to solve the optimal consensus problem for a class of discrete‐time multi‐agent systems with completely unknown dynamics. Different from the classical RL‐based optimal control algorithms based on one‐step temporal difference method, a multi‐step‐based (also call n‐step) policy gradient ADP (MS‐PGADP) algorithm, which have been proved to be more efficient owing to its faster propagation of the reward, is proposed to obtain the iterative control policies. Moreover, a novel Q‐function is defined, which estimates the performance of performing an action in the current state. Then, through the Lyapunov stability theorem and functional analysis, the proof of optimality of the performance index function is given and the stability of the error system is also proved. Furthermore, the actor‐critic neural networks are used to implement the proposed method. Inspired by deep Q network, the target network is also introduced to guarantee the stability of NNs in the process of training. Finally, two simulations are conducted to verify the effectiveness of the proposed algorithm.

Inverse Reinforcement Q-Learning Through Expert Imitation for Discrete-Time Systems.

In inverse reinforcement learning (RL), there are two agents. An expert target agent has a performance cost function and exhibits control and state behaviors to a learner. The learner agent does not know the expert's performance cost function but seeks to reconstruct it by observing the expert's behaviors and tries to imitate these behaviors optimally by its own response. In this article, we formulate an imitation problem where the optimal performance intent of a discrete-time (DT) expert target agent is unknown to a DT Learner agent. Using only the observed expert's behavior trajectory, the learner seeks to determine a cost function that yields the same optimal feedback gain as the expert's, and thus, imitates the optimal response of the expert. We develop an inverse RL approach with a new scheme to solve the behavior imitation problem. The approach consists of a cost function update based on an extension of RL policy iteration and inverse optimal control, and a control policy update based on optimal control. Then, under this scheme, we develop an inverse reinforcement Q-learning algorithm, which is an extension of RL Q-learning. This algorithm does not require any knowledge of agent dynamics. Proofs of stability, convergence, and optimality are given. A key property about the nonunique solution is also shown. Finally, simulation experiments are presented to show the effectiveness of the new approach.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Optimal Energy Management of Series Hybrid Electric Vehicles With Engine Start–Stop System

This paper develops energy management (EM) control for series hybrid electric vehicles (HEVs) that include an engine start-stop system (SSS). The objective of the control is to optimally split the energy between the sources of the powertrain and achieve fuel consumption minimization. In contrast to existing works, a fuel penalty is used to characterize more realistically SSS engine restarts, to enable more realistic design and testing of control algorithms. The paper first derives two important analytic results: a) analytic EM optimal solutions of fundamental and commonly used series HEV frameworks, and b) proof of optimality of charge sustaining operation in series HEVs. It then proposes a novel heuristic control strategy, the hysteresis power threshold strategy (HPTS), by amalgamating simple and effective control rules extracted from the suite of derived analytic EM optimal solutions. The decision parameters of the control strategy are small in number and freely tunable. The overall control performance can be fully optimized for different HEV parameters and driving cycles by a systematic tuning process, while also targeting charge sustaining operation. The performance of HPTS is evaluated and benchmarked against existing methodologies, including dynamic programming (DP) and a recently proposed state-of-the-art heuristic strategy. The results show the effectiveness and robustness of the HPTS and also indicate its potential to be used as the benchmark strategy for high fidelity HEV models, where DP is no longer applicable due to computational complexity.

Dynamic modeling and experimental validation of a low frequency piezoelectric vibration energy harvester via secondary excitation of pressured fluid

A low frequency piezoelectric energy harvester using secondary excitation of pressured fluid is proposed in this paper to scavenge low frequency vibration energy in ambient environments. This harvester was characterized by an indirect vibration-to-electric energy converter via transferring the pressure energy of pressured liquid medium. A series of simulations and experiments were conducted to prove the good performance and the feasibility of this novel structure in low-frequency high-intensity vibration. The results showed that the diameter of damping orifice, initial system backpressure and proof mass brought strong influences on the resonance frequency and output voltage. Meanwhile, according to experiments, it was found that the resonance frequency was decreased with the decreasing liquid backpressure. In this case, lowest resonance frequency of 19 Hz was achieved under the condition of 10-kg proof mass and 0.2-MPa liquid backpressure. Also, there were the optimal proof mass and backpressure to maximize the generated voltage, respectively. Moreover, in the impedance matching experiments, the maximum output power reached 1.35mW at the matching load resistance of 18kΩ.

Single machine and group scheduling with random learning rates

&lt;abstract&gt;&lt;p&gt;This study mainly considers the scheduling problems with learning effects, where the learning rate is a random variable and obeys a uniform distribution. In the first part, we introduce a single machine model with location-based learning effects. We have given the theoretical proof of the optimal solution for the five objective functions. In the second part, we study the problem with group technology. Both intra-group and inter-group have location-based learning effects, and the learning rate of intra-group jobs follows a uniform distribution. We also give the optimal ranking method and proof for the two problems proposed.&lt;/p&gt;&lt;/abstract&gt;

Российская “матрица безопасности” в Сирии

The article reviews importance of seeking new formats and modalities (political-military, social, economic, and ideological) to achieve a long-term solution to the Syrian conflict. Until now, Russia remaining a major ally of official Damascus has not fully used its potential to create favorable conditions for the post-conflict recovery of Syria. By the Syrian case, the author puts forward a hypothesis about genesis of the “security matrix” which is embodied by regiments of the Russian military police deployed across the country. In addition to routine security functions, those people – often at risk to their lives – have been performing peacekeeping, humanitarian, economic, and diplomatic tasks. Such activities including facilitation of local reconciliations create new post-conflict realities of social life. Absence of a notion of the “security matrix” in political economy and diplomacy explains the novelty of the present research. The article also contains an attempt to present both a schematic image and description of the “security matrix” as a pattern of behavior whereas multiple functions carried out by the Russian military contribute to launching national mini-dialogues, strengthening public order and security as well as to improving the humanitarian and economic situation. A SWOT-analysis serves as the optimal proof of the mentioned hypothesis. In conclusion, the author outlines continued dependency of the “security matrix” on internal and external factors. At the same time, he speaks about perspective use of the Syrian experience in formulating Russian peace-building strategies to resolve other conflicts. This in turn might contribute to strengthening Russia’s role in international relations.

Effective viscosity of semi-dilute suspensions

This review is devoted to the large-scale rheology of suspensions of rigid particles in Stokes fluid. After describing recent results on the definition of the effective viscosity of such systems in the framework of homogenization theory, we turn to our new results on the asymptotic expansion of the effective viscosity in the dilute regime. This includes a new optimal proof of Einstein’s viscosity formula for the first-order expansion, as well as the continuation of this expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: suitable renormalizations are needed and are captured by means of diagrammatic expansions.

Proof of Optimality based on Greedy Algorithm for Offline Cache Replacement Algorithm

The optimal offline cache replacement algorithm is a MIN algorithm that chooses which data item to remove when a new data item is brought from lower level of cache or main memory. The optimal offline algorithm evicts a cache item whose next request is the furthest away. For a particular program, the performance behavior of the cache memory is determined by memory and block sizes and by the nature of the replacement algorithm. Replacement algorithms, however, deserve analysis because they are based on a variety of assumptions and design considerations. This paper presents a simpler proof based on the greedy algorithm. The paper demonstrates that every ideal solution can be repeatedly transformed into the solution provided by the greedy algorithm without increasing the miss of the optimal solution, hence demonstrating that the greedy solution is providing optimality.This paper also presents a new replacement algorithm named Greedy Weight-based Cache Replacement Algorithm (GWCRA), on the basis of the Greedy algorithm and it also incorporates the weighted access-based parameter, recency and frequency. The GWCRA achieves an average speedup of 57.29% when compared to LRU and SRRIP which is 55.58% and 55.65%, respectively.

Inf-sup stability implies quasi-orthogonality

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms and leads to simple optimality proofs for the Taylor-Hood discretization of the stationary Stokes problem, a finite-element/boundary-element discretization of an unbounded transmission problem, and an adaptive time-stepping scheme for parabolic equations. The main technical tools are new stability bounds for the L U LU -factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

Computing Area-Optimal Simple Polygonizations

We consider methods for finding a simple polygon of minimum ( Min-Area ) or maximum ( Max-Area ) possible area for a given set of points in the plane. Both problems are known to be NP-hard; at the center of the recent CG Challenge, practical methods have received considerable attention. However, previous methods focused on heuristic methods, with no proof of optimality. We develop exact methods, based on a combination of geometry and integer programming. As a result, we are able to solve instances of up to \( n=25 \) points to provable optimality. While this extends the range of solvable instances by a considerable amount, it also illustrates the practical difficulty of both problem variants.