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Constrained convex optimization problems arise naturally in many real-world applications. One strategy to solve them in an approximate way is to translate them into a sequence of convex feasibility problems via the recently developed level set scheme and then solve each feasibility problem using projection methods. However, if the problem is ill-conditioned, projection methods often show zigzagging behavior and therefore converge slowly. To address this issue, we exploit the bounded perturbation resilience of the projection methods and introduce two new perturbations which avoid zigzagging behavior. The first perturbation is in the spirit of k-step methods and uses gradient information from previous iterates. The second uses the approach of surrogate constraint methods combined with relaxed, averaged projections. We apply two different projection methods in the unperturbed version, as well as the two perturbed versions, to linear feasibility problems along with nonlinear optimization problems arising from intensity-modulated radiation therapy (IMRT) treatment planning. We demonstrate that for all the considered problems the perturbations can significantly accelerate the convergence of the projection methods and hence the overall procedure of the level set scheme. For the IMRT optimization problems the perturbed projection methods found an approximate solution up to 4 times faster than the unperturbed methods while at the same time achieving objective function values which were 0.5 to 5.1% lower.

Wastewater based epidemiology has become a widely used tool for monitoring trends of concentrations of different pathogens, most notably and widespread of SARS-CoV-2. Therefore, in 2022, also in Rhineland–Palatinate, the Ministry of Science and Health has included 16 wastewater treatment sites in a surveillance program providing biweekly samples. However, the mere viral load data is subject to strong fluctuations and has limited value for political deciders on its own. Therefore, the state of Rhineland–Palatinate has commissioned the University Medical Center at Johannes Gutenberg University Mainz to conduct a representative cohort study called SentiSurv, in which an increasing number of up to 12,000 participants have been using sensitive antigen self-tests once or twice a week to test themselves for SARS-CoV-2 and report their status. This puts the state of Rhineland–Palatinate in the fortunate position of having time series of both, the viral load in wastewater and the prevalence of SARS-CoV-2 in the population. Our main contribution is a calibration study based on the data from 2023-01-08 until 2023-10-01 where we identified a scaling factor ( 0.208 ± 0.031 ) and a delay ( 5.07 ± 2.30 days) between the virus load in wastewater, normalized by the pepper mild mottle virus (PMMoV), and the prevalence recorded in the SentiSurv study. The relation is established by fitting an epidemiological model to both time series. We show how that can be used to estimate the prevalence when the cohort data is no longer available and how to use it as a forecasting instrument several weeks ahead of time. We show that the calibration and forecasting quality and the resulting factors depend strongly on how wastewater samples are normalized.

Discretization methods are commonly used for solving standard semi-infinite optimization (SIP) problems. The transfer of these methods to the case of general semi-infinite optimization (GSIP) problems is difficult due to the x-dependence of the infinite index set. On the other hand, under suitable conditions, a GSIP problem can be transformed into a SIP problem. In this paper we assume that such a transformation exists globally. However, this approach may destroy convexity in the lower level, which is very important for numerical methods. We present in this paper a solution approach for GSIP problems, which cleverly combines the above mentioned two techniques. It is shown that the convergence results for discretization methods in the case of SIP problems can be transferred to our transformation-based discretization method under suitable assumptions on the transformation. Finally, we illustrate the operation of our approach as well as its performance on several examples, including a problem of volume-maximal inscription of multiple variable bodies into a larger fixed body, which has never before been considered as a GSIP test problem.

In Rhineland-Palatinate, Germany, a system of three data sources has been established to track the Covid-19 pandemic. These sources are the number of Covid-19-related hospitalizations, the Covid-19 genecopies in wastewater, and the prevalence derived from a cohort study. This paper presents an extensive comparison of these parameters. It is investigated whether wastewater data and a cohort study can be valid surrogate parameters for the number of hospitalizations and thus serve as predictors for coming Covid-19 waves. We observe that this is possible in general for the cohort study prevalence, while the wastewater data suffer from a too large variability to make quantitative predictions by a purely data-driven approach. However, the wastewater data and the cohort study prevalence are able to detect hospitalizations waves in a qualitative manner. Furthermore, a detailed comparison of different normalization techniques of wastewater data is provided.

In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility problems by iteratively constraining the objective function from above until the feasibility problem is inconsistent. For each of the feasibility problems one may apply any of the existing projection methods for solving it. In particular, the scheme allows the use of subgradient projections and does not require exact projections onto the constraints sets as in existing similar methods. We also apply the newly introduced concept of superiorization to optimization formulation and compare its performance to our scheme. We provide some numerical results for convex quadratic test problems as well as for real-life optimization problems coming from medical treatment planning.

Urgency of fresh items is increasing day by day everywhere. During transportation over a long distance, the rate of deterioration is also increasing for perishable items, and if the items are not reaching at destination within a specified time, then they are no longer fresh. Based on the fact, the customers loose the quality and quantity of items; as a result, suppliers are penalized for the occurrence of any such a situation. This problem is resolved by introducing two criteria such as time window restrictions and preservation technology in the proposed study. To prevent the deterioration rate, we find a strategy by formulating a model for multi-objective fixed-charge solid transportation problem with two criteria and then finalize one of them. To ensure that the necessary fresh items are to reached at the destination just in time, we introduce several objectives such as transportation cost, transportation time, preservation cost, penalty charge for time window, carbon emission with cap policy, deterioration, etc., such that these are optimized simultaneously. Again source, demand and conveyance capacity are not precisely estimated for different real-life situations. Here, we address such situation by choosing type-2 zigzag uncertain variable. Expected value operator is initiated to convert such uncertainty into crisp form, and then three approaches, namely, fuzzy programming, Pythagorean hesitant fuzzy programming and global criterion method are utilized to find Pareto-optimal solution. Finally two numerical examples are put to check the appropriateness of the formulated study. The results along with discussions and conclusions are described at last. The main contribution is that the deterioration of transported perishable items is reduced by imposing one of the proposed criteria according to the economical or other conditions of systems.

In this article we introduce an algorithm that approximates the nondominated sets of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches for problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which are multiobjective convex problems, by fixing the integer assignments. The patch problems are solved using (simplicial) Sandwiching. We identify parts of patches that are dominated by other patches and ensure that these patch parts are not refined further. We prove that the algorithm converges and show a bound on the reduction of the approximation error in the course of the algorithm. We illustrate the behaviour of our algorithm using some numerical examples and compare its performance to an algorithm from literature.

Waste management involved in various fields of global ecosystem that provides several positive effects for green environment and sustainable development. We devise a multi-objective solid transportation model of waste management problem in agriculture field and forest department for urban or rural development. Starting to end point of the problem covered by considering the objective functions as transportation cost, job opportunity and carbon emission. Carbon emission is restricted by the combination of several policies of carbon mechanism (carbon tax, cap-and-trade and offset policy). Various critical sitchs appear in such realistic process and uncertainty attached with related data. Here we prefer Pythagorean hesitant fuzzy environment to overcome deep uncertainty rather than single uncertainty. After that, we initiate a ranking approach to convert uncertain data into crisp data. To justify the appropriateness of the formulated model and to select the best policy of carbon mechanism, we study two industrial applications with various cases of such mechanism. To derive the Pareto-optimal solution of the problems, two fuzzy techniques, namely, fuzzy programming and Pythagorean hesitant fuzzy programming, are utilized here. Comparative study, model validation, sensitivity analysis, managerial insights and conclusions with future research scopes are outlined at last.

Modelling real processes often results in several suitable models. In order to be able to distinguish, or discriminate, which model best represents a phenomenon, one is interested, e.g., in so-called T-optimal designs. These consist of the (design) points from a generally continuous design space at which the models deviate most from each other under the condition that they are best fitted to those points. Thus, the T-criterion represents a bi-level optimization problem, which can be transferred into a semi-infinite one but whose solution is very unstable or time consuming for non-linear models and non-convex lower- and upper-level problems. If one considers only a finite number of possible design points, a numerically well tractable linear semi-infinite optimization problem arises. Since this is only an approximation of the original model discrimination problem, we propose an algorithm which alternately and adaptively refines discretizations of the parameter as well as of the design space and, thus, solves a sequence of linear semi-infinite programs. We prove convergence of our method and its subroutine and show on the basis of discrimination tasks from process engineering that our approach is stable and can outperform the known methods.

In this paper we study the possibilities of sharing profit in combinatorial procurement auctions and exchanges. Bundles of heterogeneous items are offered by the sellers, and the buyers can then place bundle bids on sets of these items. That way, both sellers and buyers can express synergies between items and avoid the well-known risk of exposure (see, e.g., Cramton, Shoham, & Steinberg, Combinatorial Auctions, 2006 ). The reassignment of items to participants is known as the Winner Determination Problem (WDP). We propose solving the WDP by using a Set Covering formulation, because profits are potentially higher than with the usual Set Partitioning formulation, and subsidies are unnecessary. The achieved benefit is then to be distributed amongst the participants of the auction, a process which is known as profit sharing. The literature on profit sharing provides various desirable criteria. We focus on three main properties we would like to guarantee: Budget balance, meaning that no more money is distributed than profit was generated, no negative transfers, which guarantees to each player that participation does not lead to a loss, and the core property, which provides every subcoalition with enough money to keep them from separating. We characterize all profit distributions that satisfy these three conditions by a monetary flow network and establish a connection to the famous VCG payment scheme (Clarke in Public Choice 18:19ff, 1971 ; Groves in Econometrica 41:617–631, 1973 ; Vickrey in J. Finance 16:8–37, 1961 ) and the Shapley Value (Shapley in Kuhn & Tucker (Eds.), Contributions to the Theory of Games, vol. II, pp. 307–317, 1953 ). Finally, we introduce a novel profit sharing scheme based on scaling down VCG discounts using max-min fairness principles to achieve budget balance. It approximates the Shapley Value linearly in the number of bundles and can be computed efficiently.