Non-convex Research Articles

On an abrasion-motivated fractal model

Abstract In this paper, we consider a fractal model motivated by the abrasion of convex polyhedra, where the abrasion is realised by chipping small neighbourhoods of vertices. After providing a formal description of the successive chippings, we show that the net of edges converge to a compact limit set under mild assumptions. Furthermore, we study the upper box-counting dimension and the Hausdorff dimension of the limiting object of the net of edges after infinitely many chipping.

Properties of generalized polyhedral convex multifunctions

This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges of generalized polyhedral convex multifunctions, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results provide a framework for representing the relative interior of the graph of a generalized polyhedral convex multifunction in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex multifunctions from Banach spaces to locally convex topological vector spaces.

Quad-Rotor Avoidance Trajectory Generation for Convex Polyhedron Obstacles

This study investigates a method for generating obstacle avoidance trajectories for arbitrary convex polyhedrons. We propose a formulation that converts discrete conditions into continuous equation forms to avoid convex polyhedron obstacles. The condition that the evaluation point be located outside of the convex polyhedron can be transformed into a constraint in the continuous equation form and incorporated into the optimization calculation to generate avoidance trajectories. Avoidance trajectory generation using the Legendre pseudospectral method is performed for convex polyhedral obstacles of various shapes. The results show that the proposed method successfully generates avoidance trajectories for arbitrary convex polyhedral obstacles.

Design and tension distribution optimization of a 9-DOF cable-driven parallel spray-painting robot with 3 degrees of redundancy

This paper presents a 9-DOF cable-driven parallel spray-painting robot (CDPSR) with 3 ° of redundancy (DORs) for automated spraying on large, curved surfaces and proposes a feasible cable tension region (FCTR) calculation algorithm for multi-redundant robots. The end effector of the CDPSR consists of two moving platforms connected by a spherical hinge and driven by 12 cables, enabling 9 DOFs of motion (3 for translation and 6 for rotation). Given that the CDPSR has three DORs, a multi-dimensional FCTR vertex calculation algorithm is introduced. The FCTR is an adjacent, continuous, and closed convex hull in every dimension; therefore, the vertices of the FCTR can be determined sequentially and dimensionally. For DOR=3, the vertices on one face of the convex polyhedron are first determined and then extended to adjacent faces until the polyhedron is closed. An experimental prototype of the proposed robot was constructed and experiments on spray trajectory planning and FCTR calculation validation were conducted. The results of the simulations and experiments verified the motion performance of the CDPSR and the accuracy and efficiency of the FCTR calculation algorithm.

Optimization of renewable energy-based integrated energy systems: A three-stage stochastic robust model

The integration of renewable energy into an integrated energy system (REIES) represents a promising approach to achieving clean and low-carbon energy consumption. However, the inherent uncertainty and variability of renewable energy sources and load demands present significant challenges to the operational efficiency and stability of REIES. To address these challenges, we propose a three-stage stochastic robust optimization (TRSRO) model. This model accounts for market electricity prices, source-load scenario probabilities, and output uncertainties to optimize system configurations in REIES. It also includes the optimization of power exchanges with the regional grid and strategic operational scheduling. Initial scenarios of uncertainty variables are generated using the spectral normalized conditional Wasserstein generative adversarial network (SNCWGAN), forming polyhedral uncertainty sets. Comprehensive norm constraints are then applied to capture probability distribution uncertainties within these sets. To improve the efficiency of solving the model, we introduce a two-layer column constraint generation algorithm, considering variable alternate iterations (TLCCG-VAI). The results demonstrate that the proposed method can achieve a 12.16 % reduction in total cost compared to the baseline method. Furthermore, the TLCCG-VAI algorithm reduces runtime by 82.97 % while maintaining the same level of solution accuracy.

Robust wind power capacity planning under fuel price uncertainty using conic duality theory and piecewise McCormick relaxation

Increasing the proportion of renewable energy sources (RESs) in power generation is crucial due to fossil fuel depletion and rising environmental pollution. In this regard, identifying the most lucrative sites and sizes for installing wind farms, as one of the fastest-growing types of RESs, is imperative. This paper presents a robust profit-oriented wind power capacity planning (WPCP) considering long- and short-term uncertainty. The model forms a two-stage min-max-min hierarchical structure. The first stage minimizes the investment cost plus maximum regret, while the second stage maximizes the profit under the worst-case uncertainty realization. Unlike the existing approaches where uncertainty in fossil fuel prices is neglected, we model fuel price uncertainty using both polyhedral and ellipsoidal uncertainty sets. In this respect, the third level is formulated as a bi-level program, with the upper level being the profit maximization and the lower level being the locational marginal price (LMP) calculation. In the case of the ellipsoidal set, the conic duality theory is employed to dualize the lower level. The piecewise McCormick relaxation (PMR) technique linearizes the bilinear terms. The nested column-and-constraint generation (NCCG) technique solves the formulated problem. A clarifying case study is employed to demonstrate the efficacy of the proposed model.

A multi-objective robust dispatch strategy for renewable energy microgrids considering multiple uncertainties

The demand for low-carbon transformations and the uncertainty of renewable energy sources and loads present significant challenges for the optimal dispatch of microgrid. This study proposed a multi-objective robust dispatch strategy to reduce the risks associated with the uncertainty of renewable energy source output and loads while promoting low-carbon and economical microgrid operation. The economic emission dispatch problem for a microgrid was formulated as a multi-objective robust dual-layer optimization model. Consequently, a high-dimensional adjustable linear polyhedral uncertainty set was proposed to describe the uncertainty of renewable energy sources and loads. This study transformed the original model into an easy-to-solve single-layer second-order cone programming optimal power flow optimization model by employing second-order cone relaxation and duality transformation. Thereafter, a synthetic membership function was proposed to determine the optimal compromise solution. To determine the charging and discharging statuses of the battery storage system and the electricity traded between the microgrid and the external power grid, a battery storage system control strategy based on time-of-use electricity prices and real-time power flow calculations was proposed. Simulations conducted on a modified IEEE-30 bus system demonstrated that the proposed strategy effectively reduced the economic costs and carbon emissions of the microgrid by 8.23 % and 2.43 %, respectively.

Robust convoy movement problem under travel time uncertainty

A convoy represents a collection of vehicles traveling with a spacing of 50–100 m between them for tactical purposes. The convoy movement problem is a variant of the vehicle routing problem, an NP-hard problem aimed at determining the paths and schedules of convoys. Given the uncertainties in travel times during wartime, attributable to various factors such as road conditions and enemy threats, it is essential to consider uncertain travel times when determining convoy paths and schedules. Therefore, this study introduces a robust convoy movement problem under travel time uncertainty. A polyhedral set for uncertain travel times is used to derive a robust counterpart for the problem. To solve the proposed problem, we establish an exact algorithm that determines optimal solutions by iteratively generating and integrating multiple paths of convoys. This algorithm involves four steps: generation of k-th robust shortest paths, construction of path combinations, adjustment of departure times, and conduction of optimality check. These steps are iterated sequentially until the optimal solution is obtained. In computational experiments, the exact algorithm demonstrates superior performance and reduced computation time compared with the commercial solver CPLEX on both real instances and randomly generated instances. In addition, we conduct a sensitivity analysis for several parameters related to the problem, providing valuable managerial insights for decision-makers.

Fuzzy-stochastic distributional robust approach for microalgae-based bio-plastic supply chain with new sustainability aspects

The current study offers a sustainable supply chain model that can be used to identify opportunities and constraints for the future of the microalgae-based bio-diesel and bio-plastic industry. The network’s carbon emissions are restricted by an uncertain constraint that allows for flexibility in controlling emissions as needed. The social constraints of the model provides a transparent depiction of labour needs in dynamic business landscape and foster women’s empowerment through a preference coefficient. The proposed model introduces a new upside risk measures under fuzzy-stochastic setting to quantify the upward financial risk. To characterize and manage the model uncertainty, a fuzzy-stochastic distributional robust model is presented. Credibility distribution is utilized to handle fuzzy uncertainty, while box and polyhedral uncertainty sets are included to handle stochastic uncertainty. After that, tractable deterministic formulations of the uncertain model for two ambiguity sets are presented to solve the model. Finally, a number of insights are obtained by means of five test problems that are implemented in Python and resolved by one exact and five meta-heuristic algorithms. Experiments indicate that, when faced with uncertainty, two robust models have produced more stable solutions (16.23% and 19.67% higher stability, respectively) than nominal stochastic model. Further, a 4.68% decline in total cost and 3.27% increment in bio-diesel production can be achieved by increasing bio-diesel conversion rate to 10%. In addition, a 9.09% downturn in total cost and 10.44% rise in bio-plastic production can be obtained by 10% increment in bio-plastic conversion rate. Moreover, 45.38% more woman labours can be recruited by increasing the preference coefficient from −0.2 to +0.2.

Exact methods and a variable neighborhood search for the robust capacitated [formula omitted]-median problem

The capacitated p-median problem (CPMP) involves placing p identical facilities in a network and assigning customer nodes to these facilities to satisfy all customer demands with minimal transportation costs. In practical applications, demand and distance parameters are often uncertain during the planning process, leading to infeasible or excessively costly solutions if these uncertainties are disregarded. This paper addresses the robust CPMP (RCPMP), which incorporates demand uncertainty into the problem using the robust optimization paradigm. We propose a general framework to model and solve the RCPMP, considering different polyhedral uncertainty sets, namely the cardinality-constrained and the knapsack sets. We develop exact approaches encompassing compact models, different families of valid inequalities, and branch-and-cut and branch-and-price algorithms, exploring both implemented uncertainty sets and problem structure. Furthermore, we implement an efficient Variable Neighborhood Search (VNS) heuristic to solve these robust variants, which incorporates state-of-the-art algorithms, parallelization techniques, and optimized data structures. Computational experiments using adapted benchmark instances with up to 400 nodes indicate the effectiveness of the proposed approaches. The results show that using parallelization and hash tables within the VNS heuristic promotes significant performance improvements and yields near-optimal solutions for most instances, as well as outperforming the exact approaches in several instances where the optimal solution was not found. Moreover, these results highlight the benefits of using robust solutions in practical settings, especially when considering different uncertainty sets to generate solutions with advantageous trade-offs between cost and risk.

Estimation of the error in determining the motion parameters of maneuvering objects

The study contains the definition of errors in estimating the motion parameters of maneuvering objects that arise due to the discrepancy between the real model and the model used to build the filter. In the real model, along with noise and stochastic uncertainties, non-stochastic uncertainties act. The solution to the problem of estimation errors is based on the use of convex polyhedra. The statement of the research problem is given. It is shown that since the observer uses the motion model and the measurement equation, the source of estimation errors is not only the above, but also the fact that along with the random component of the measurement noise, there is also a non-stochastic component that is not taken into account by the observer. At the same time, the statistical characteristics of the latter do not exist or are unknown. It is shown that the use of a filter, for example, Kalman, will lead to errors. The conditions that determine the average filtering error and the total standard error of the estimation are described. It is noted that the obtained calculation results allow us to estimate the maximum permissible values used in the design of the filter for the most unfavorable case of evaluation. The relationship between the filtration quality and the parameters of the effects is determined. Expressions are obtained regarding the filtration quality for the worst and the best for object maintenance. The asymptotic properties of filtering errors are considered. It is shown that the "convergence" of the filter means its final "memory", since errors at the next step do not depend on errors from all previous steps, but only on steps from a certain value. The conditions under which the potential accuracy of the filter is achieved are also determined. The solution of extreme problems is described. You can determine the maximum filtering quality by iterating over the values on a set of corner points or using extremum conditions. Numerical simulation of extreme values of filtering errors is given. The asymptotic properties of filtering errors are considered. It is shown that the "convergence" of the filter means its final "memory", since errors at the next step do not depend on errors from all previous steps, but only on steps from a certain value. The conditions under which the potential accuracy of the filter is achieved are also determined. The solution of extreme problems is described. You can determine the maximum filtering quality by iterating over the values on a set of corner points or using extremum conditions. Numerical simulation of extreme values of filtering errors is given. Expressions for the set of average values of filtering errors are found, asymptotic and extreme properties of errors and sets are considered. It is shown that the conducted analysis of the accuracy of the filtering algorithm is evaluative in nature and is designed for the worst case. If the accuracy is not satisfactory, it is necessary to either change the model or synthesize a minimax-stochastic filter.

Two-Stage Robust Optimization of Integrated Energy Systems Considering Uncertainty in Carbon Source Load

Integrated Energy Systems (IESs) interconnect various energy networks to achieve coordinated planning and optimized operation among heterogeneous energy subsystems, making them a hot topic in current energy research. However, with the high integration of renewable energy sources, their fluctuation characteristics introduce uncertainties to the entire system, including the corresponding indirect carbon emissions from electricity. To address these issues, this paper constructs a two-stage, three-layer robust optimization operation model for IESs from day-ahead to intra-day. The model analyzes the uncertainties in carbon emission intensity at grid-connected nodes, as well as the uncertainty characteristics of photovoltaic, wind turbine, and cooling, heating, and electricity loads, expressed using polyhedral uncertainty sets. It standardizes the modeling of internal equipment in the IES, introduces carbon emission trading mechanisms, and constructs a low-carbon economic model, transforming the objective function and constraints into a compact form. The column-and-constraint generation algorithm is applied to transform the three-layer model into a single-layer main problem and a two-layer subproblem for iterative solution. The Karush–Kuhn–Tucker (KKT) condition is used to convert the two-layer subproblem into a linear programming model. A case study conducted on a park shows that while the introduction of uncertainty optimization increases system costs and carbon emissions compared to deterministic optimization, the scheduling strategy is more stable, significantly reducing the impact of uncertainties on the system. Moreover, the proposed strategy reduces total costs by 5.03% and carbon emissions by 1.25% compared to scenarios considering only source load uncertainty, fully verifying that the proposed method improves the economic and low-carbon performance of the system.

Data-driven robust optimisation of hydrogen infrastructure planning under demand uncertainty using a hybrid decomposition method

In the race towards “Net-zero”, hydrogen has emerged as one of the key alternatives to carbon-based fossil fuels for a sustainable decarbonisation. This work studies the spatially explicit multi-period hydrogen infrastructure planning under demand uncertainty that contributes to the heat decarbonisation in Great Britain. Demand uncertainty surrounding future hydrogen supply chains poses challenges to cost optimisation and system security, so uncertainty-resilient policies are required to ensure robust operations. In this work, we employ data-driven robust optimisation to develop a framework for uncertainty-aware representative days explicitly characterised by polyhedral uncertainty sets. The proposed framework is applied on a multi-period mixed-integer linear model with dual temporal resolution which aims to determine the optimal yearly investment decisions and hourly operational decisions for the hydrogen infrastructure planning under demand uncertainty. To efficiently solve the large-scale two-stage adaptive robust optimisation problem, a hybrid decomposition algorithm is developed based on a two-step hierarchical procedure and the column-and-constraint generation method, which can significantly reduce the computational complexity. The optimisation results highlight how uncertainty can result in the total cost increase, and verify the advantages on controlling solution conservatism in the adaptive robust optimisation compared to the static robust optimisation.

Speeding up static analysis with the split operator

In the context of abstract interpretation-based static analysis, we propose a new abstract operator modeling the split of control flow paths: the goal of the operator is to enable a more efficient analysis when using abstract domains that are computationally expensive, having no negative effect on precision, and occasionally resulting in a more precise analysis. We focus on the case of conditional branches guarded by numeric linear constraints, including implicit numerical branches. We provide an experimental evaluation of real-world test cases, showing that by using the split operator we can achieve significant efficiency improvements with respect to the classical approach for a static analysis based on the domain of convex polyhedra. We also briefly discuss the applicability of this new operator to different, possibly non-numeric abstract domains.

The natural ordering cone of a polyhedral convex set-valued objective mapping

ABSTRACT For a given polyhedral convex set-valued mapping we define a polyhedral convex cone which we call the natural ordering cone. We show that the solution behaviour of a polyhedral convex set optimization problem can be characterized by this cone. Under appropriate assumptions, the natural ordering cone is proven to be the smallest ordering cone which makes a polyhedral convex set optimization problem solvable.

Research on motion planning system for wall-climbing mobile manipulator for large steel structures welding operation

Purpose The purpose of this study is to address the welding demands within large steel structures by presenting a global spatial motion planning algorithm for a mobile manipulator. This algorithm is based on an independently developed wall-climbing robot, which comprises a four-wheeled climbing mobile platform and a six-degree-of-freedom robotic manipulator, ensuring high mobility and operational flexibility. Design/methodology/approach A convex hull feasible domain constraint is developed for motion planning in the mobile manipulator. For extensive spatial movements, connected sequences of convex polyhedra are established between the composite robot’s initial and target states. The composite robot’s path and obstacle avoidance optimization problem are solved by constraining the control points on B-spline curves. A dynamic spatial constraint rapidlye-xploring random trees-connect (RRTC) motion planning algorithm is proposed for the manipulator, which quickly generates reference paths using spherical spatial constraints at the manipulator’s end, eliminating the need for complex nonconvex constraint modeling. Findings Experimental results show that the proposed motion planning algorithm achieves optimal paths that meet task constraints, significantly reducing computation times in task conditions and shortening operation times in non-task conditions. Originality/value The algorithm proposed in this paper holds certain application value for the realization of automated welding operations within large steel structures using mobile manipulator.

Spotting, Tracking algorithm and the remoteness

On this paper we present a solution to detect and know if a point M is inside a polygon ( A(k), k∈{1,...,n} ) or outside. We are going to give a very simple, practical and explicit method of the triangulation of a convex polygon (convex polyhedron) after a definition and the concretization of the order relation of the points of a polygon in a plane following a well-chosen orientation in before and an arbitrary point of the vertices of the polygon. In the case where the point M is outside the polygon, a simple optimization method will be applied to determine the distance between the point M and the polygon A(1), ..., A(n) and the point P of the border of the polygon closest to M ”The neighboring Point”.

On the Convex Formulations of Robust Markov Decision Processes

Robust Markov decision processes (MDPs) are used for applications of dynamic optimization in uncertain environments and have been studied extensively. Many of the main properties and algorithms of MDPs, such as value iteration and policy iteration, extend directly to RMDPs. Surprisingly, there is no known analog of the MDP convex optimization formulation for solving RMDPs. This work describes the first convex optimization formulation of RMDPs under the classical sa-rectangularity and s-rectangularity assumptions. By using entropic regularization and exponential change of variables, we derive a convex formulation with a number of variables and constraints polynomial in the number of states and actions, but with large coefficients in the constraints. We further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or entropy-based uncertainty sets, showing that, in these cases, RMDPs can be reformulated as conic programs based on exponential cones, quadratic cones, and nonnegative orthants. Our work opens a new research direction for RMDPs and can serve as a first step toward obtaining a tractable convex formulation of RMDPs. Funding: The work in the paper was supported, in part, by NSF [Grants 2144601 and 1815275]; and Agence Nationale de la Recherche [Grant 11-LABX-0047].

Dynamic shear behavior of CFRP-concrete interface: Test and 3D mesoscale numerical simulation

The study on dynamic shear behavior of fiber reinforced polymer (FRP)-concrete interface is of great significance for the performance evaluation and design of FRP externally strengthened concrete structures. Firstly, the quasi-static and dynamic single shear test on the interfacial shear behavior between carbon fiber reinforced polymer (CFRP) sheet and concrete substrate was carried out. The corresponding failure modes of CFRP-concrete interface, CFRP strain-time histories, load-displacement curves, and interfacial shear stress-slip relationships under loading rates of 8.33 × 10−6–10 m·s−1 were obtained. It was indicated that the dynamic interfacial shear behaviors, i.e., interfacial failure mode, debonding load, interfacial shear stress, etc., were sensitive to loading rates. Then, a 3D mesoscale modeling approach of concrete with random shaped, sized, and spatially distributed convex polyhedron aggregates was proposed, in which the volume fraction of aggregates was adjustable within the range of 0–50 % through gravitational drop and size scaling of aggregates. Furthermore, based on the established 3D mesoscale concrete model and the zero-thickness cohesive elements for adhesive layer, numerical simulations for FPR-concrete interfacial shear behavior were conducted and validated by comparing with the present and existing quasit-static and dynamic single shear tests. The experimental phenomenon of the failure location transferring from concrete substrate to adhesive layer at high loading rates was numerically reproduced. Finally, the influences of strain rate enhancing effects of aggregates and mortar on the interfacial failure modes were discussed. It was revealed that the failure location transferring from the concrete substrate to the adhesive layer was significantly affected by the strength enhancement of mortar and aggregates with the loading rate increasing.

On Multilinear Inequalities of Ho ̈lder-Brascamp-Lieb Type for Torsion-Free Discrete Abelian Groups

Holder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of L^p norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequalities for torsion-free discrete Abelian groups in three respects.The optimal constant in any such inequality is shown to equal 1 whenever it is finite.An algorithm that computes the admissible polyhedron of exponents is developed. It is shown that nonetheless, existence of an algorithm that computes the full list of inequalitiesin the Bennett-Carbery-Christ-Tao description of the admissible polyhedron for all data,is equivalent to an affirmative solution of Hilbert's Tenth Problem over the rationals.That problem remains open.