Constraint Optimization Problem Research Articles

Dynamic ε-multilevel hierarchy constraint optimization with adaptive boundary constraint handling technology

Real-world optimization problems are often difficult to solve because of the complexity of the objective function and the large number of constraints that accompany it. To solve such problems, we propose Adaptive Dynamic ε-Multilevel Hierarchy Constraint Optimization (εMHCO). Firstly, we propose the dynamic constraint tolerance factor ε which can change dynamically with the feasible ratio and the number of iterations in the current population. This ensures a reasonable proportion of virtual feasible solutions in the population. Secondly, we propose adaptive boundary constraint handling technology (ABCHT). It can reshape the current individual position adaptively according to the size of constraint violation and increase the diversity of the population. Finally, we propose multi-level hierarchy optimization, whose multiple population structure is beneficial to solve real-world constraint optimization problems (COPs). To validate and analyze the performance of εMHCO, numerical experiments are conducted on the latest real-world test suite CEC’2020, which contains a set of 57 real-world COPs, and compared with four state-of-the-art algorithms. The results show that εMHCO is significantly superior to, or at least comparable to the state-of-the-art algorithms in solving real-world COPs. Meanwhile, the effectiveness and feasibility of εMHCO are verified on the real-world problem of the pipeline inner detector speed control.

Bald Eagle Search Optimization Algorithm Combined with Spherical Random Shrinkage Mechanism and Its Application

Over the last two decades, stochastic optimization algorithms have proved to be a very promising approach to solving a variety of complex optimization problems. Bald eagle search optimization (BES) as a new stochastic optimization algorithm with fast convergence speed has the ability of prominent optimization and the defect of collapsing in the local best. To avoid BES collapse at local optima, inspired by the fact that the volume of the sphere is the largest when the surface area is certain, an improved bald eagle search optimization algorithm (INMBES) integrating the random shrinkage mechanism of the sphere is proposed. Firstly, the INMBES embeds spherical coordinates to design a more accurate parameter update method to modify the coverage and dispersion of the population. Secondly, the population splits into elite and non-elite groups and the Bernoulli chaos is applied to elite group to tap around potential solutions of the INMBES. The non-elite group is redistributed again and the Nelder-Mead simplex strategy is applied to each group to accelerate the evolution of the worst individual and the convergence process of the INMBES. The results of Friedman and Wilcoxon rank sum tests of CEC2017 in 10, 30, 50, and 100 dimensions numerical optimization confirm that the INMBES has superior performance in convergence accuracy and avoiding falling into local optimization compared with other potential improved algorithms but inferior to the champion algorithm and ranking third. The three engineering constraint optimization problems and 26 real world problems and the problem of extracting the best feature subset by encapsulated feature selection method verify that the INMBES’s performance ranks first and has achieved satisfactory accuracy in solving practical problems.

Fast DOA Estimation Algorithms via Positive Incremental Modified Cholesky Decomposition for Augmented Coprime Array Sensors.

This paper proposes a fast direction of arrival (DOA) estimation method based on positive incremental modified Cholesky decomposition atomic norm minimization (PI-CANM) for augmented coprime array sensors. The approach incorporates coprime sampling on the augmented array to generate a non-uniform, discontinuous virtual array. It then utilizes interpolation to convert this into a uniform, continuous virtual array. Based on this, the problem of DOA estimation is equivalently formulated as a gridless optimization problem, which is solved via atomic norm minimization to reconstruct a Hermitian Toeplitz covariance matrix. Furthermore, by positive incremental modified Cholesky decomposition, the covariance matrix is transformed from positive semi-definite to positive definite, which simplifies the constraint of optimization problem and reduces the complexity of the solution. Finally, the Multiple Signal Classification method is utilized to carry out statistical signal processing on the reconstructed covariance matrix, yielding initial DOA angle estimates. Experimental outcomes highlight that the PI-CANM algorithm surpasses other algorithms in estimation accuracy, demonstrating stability in difficult circumstances such as low signal-to-noise ratios and limited snapshots. Additionally, it boasts an impressive computational speed. This method enhances both the accuracy and computational efficiency of DOA estimation, showing potential for broad applicability.

Exploiting Functional Constraints in Automatic Dominance Breaking for Constraint Optimization

Dominance breaking is a powerful technique in improving the solving efficiency of Constraint Optimization Problems (COPs) by removing provably suboptimal solutions with additional constraints. While dominance breaking is effective in a range of practical problems, it is usually problem specific and requires human insights into problem structures to come up with correct dominance breaking constraints. Recently, a framework is proposed to generate nogood constraints automatically for dominance breaking, which formulates nogood generation as solving auxiliary Constraint Satisfaction Problems (CSPs). However, the framework uses a pattern matching approach to synthesize the auxiliary generation CSPs from the specific forms of objectives and constraints in target COPs, and is only applicable to a limited class of COPs. This paper proposes a novel rewriting system to derive constraints for the auxiliary generation CSPs automatically from COPs with nested function calls, significantly generalizing the original framework. In particular, the rewriting system exploits functional constraints flattened from nested functions in a high-level modeling language. To generate more effective dominance breaking nogoods and derive more relaxed constraints in generation CSPs, we further characterize how to extend the system with rewriting rules exploiting function properties, such as monotonicity, commutativity, and associativity, for specific functional constraints. Experimentation shows significant runtime speedup using the dominance breaking nogoods generated by our proposed method. Studying patterns of generated nogoods also demonstrates that our proposal can reveal dominance relations in the literature and discover new dominance relations on problems with ineffective or no known dominance breaking constraints.

Scheduling operations in a large hospital by multiple agents

The scheduling of operations in a large hospital is performed jointly by several groups of people, each with its own objective and constraints. It is a two-phase process, starting with the allocation of operating rooms to wards, and followed by the scheduling of operations in each operating room of the hospital on each day. The final schedule must satisfy all inter-ward hard constraints, such as the allocation of anesthetists, nurses, and equipment to operations that are taking place in parallel, and ideally, it should also address soft constraints such as taking the urgency and complexity of operations into consideration.This study contributes to the ongoing effort of adapting multi-agent optimization models and algorithms to real-world applications by modeling the problems in both phases as distributed constraint optimization problems (DCOPs), with different properties. The first phase includes partially cooperative ward-representing agents, allocating operating rooms for daily usage among themselves. In the second phase, ward-representing agents interact with agents representing constraining elements, in order to generate daily operation schedules for each operating room, thus forming a unique bipartite constraint graph. On one side are the ward representatives, while on the other are the agents representing the constraining resources. Each agent has a non-trivial local problem to solve, and its solution serves as the proposed assignment in the distributed algorithm.The study begins by discussing the properties required of the algorithms needed to solve the two phases. It then proposes adjustments to existing distributed partially cooperative algorithms and local search algorithms to solve these problems, and compares the results of different variants of these algorithms. The results obtained for both phases emphasize that successful collaboration is predicated on two requirements: that agents hold consistent information regarding their peers’ states and that the degree of exploration undertaken by the algorithm is restricted in order to produce high-quality solutions.

Proposed residual stress model for hot‐rolled wide flange steel cross sections

AbstractResidual stresses in hot‐rolled wide flange steel cross sections may lead to premature yielding, accelerated corrosion and brittle fracture of steel members. The above phenomena lead to a loss of a steel member's stiffness and resistance under mechanical loading. Available residual stress models are mostly based on residual stress measurements that date back to 1950s. Some of the drawbacks of these models relate to the lack of consistency in the considered parameters that affect the residual stress development within a steel cross section. Motivated by this, this paper proposes a new residual stress model for hot‐rolled wide flange steel cross sections. The proposed residual stress model relies on a dataset of 80 experiments that are complemented by additional measurements done as part of the present study. A parabolic residual stress distribution, which is deduced from a constraint optimization problem, is fitted to the assembled data. The proposed residual stress distributions are generalized with the aid of rigorous statistical analyses. The results suggest that the residual stress model is highly dependent on the cross‐sectional area and the depth‐to‐width ratio of a hot‐rolled wide flange steel cross section. The proposed residual stress model reduces the error, on average, by 60‐70% compared to available residual stress models (e.g., European Convention for Constructional Steelwork model), regardless of the cross‐sectional geometry of the hot‐rolled cross section.

Learning to select SAT encodings for pseudo-Boolean and linear integer constraints

Many constraint satisfaction and optimisation problems can be solved effectively by encoding them as instances of the Boolean Satisfiability problem (SAT). However, even the simplest types of constraints have many encodings in the literature with widely varying performance, and the problem of selecting suitable encodings for a given problem instance is not trivial. We explore the problem of selecting encodings for pseudo-Boolean and linear constraints using a supervised machine learning approach. We show that it is possible to select encodings effectively using a standard set of features for constraint problems; however we obtain better performance with a new set of features specifically designed for the pseudo-Boolean and linear constraints. In fact, we achieve good results when selecting encodings for unseen problem classes. Our results compare favourably to AutoFolio when using the same feature set. We discuss the relative importance of instance features to the task of selecting the best encodings, and compare several variations of the machine learning method.

Study on reservoir optimal operation based on coupled adaptive ε constraint and multi strategy improved Pelican algorithm

The optimal operation of reservoir groups is a strongly constrained, multi-stage, and high-dimensional optimization problem. In response to this issue, this article couples the standard Pelican optimization algorithm with adaptive ε constraint methods, and further improves the optimization performance of the algorithm by initializing the population with a good point set, reverse differential evolution, and optimal individual t-distribution perturbation strategy. Based on this, an improved Pelican algorithm coupled with adaptive ε constraint method is proposed (ε-IPOA). The performance of the algorithm was tested through 24 constraint testing functions to find the optimal ability and solve constraint optimization problems. The results showed that the algorithm has strong optimization ability and stable performance. In this paper, we select Sanmenxia and Xiaolangdi reservoirs as the research objects, establish the maximum peak-cutting model of terrace reservoirs, apply the ε-IPOA algorithm to solve the model, and compare it with the ε-POA (Pelican algorithm coupled with adaptive ε constraint method) and ε-DE (Differential Evolution Algorithm) algorithms, the results indicate that ε. The peak flow rate of the Huayuankou control point solved by the IPOA algorithm is 12,319 m3/s, which is much lower than the safe overflow flow rate of 22,000 m3/s at the Huayuankou control point, with a peak shaving rate of 44%, and other algorithms do not find effective solutions meeting the constraint conditions. This paper provides a new idea for solving the problem of flood control optimal operation of cascade reservoirs.

Energy-Efficient Power Allocation for Full-Duplex Device-to-Device Underlaying Cellular Networks with NOMA

Full-duplex (FD), Device-to-Device (D2D) and non-orthogonal multiple access (NOMA) are promising wireless communication techniques to improve the utilization of spectrum resources. Meanwhile, introducing FD, D2D and NOMA in cellular networks is very challenging due to the complex interference problem. To deal with the complex interference of FD D2D underlaying NOMA cellular networks, power allocation (PA) is extensively studied as an efficient interference management technique. However, most of the previous research works on PA to optimize energy efficiency only consider the system framework of partially joint combining techniques of FD, D2D and NOMA, and the constraints of optimization problem are very different. In this paper, in order to further improve the energy efficiency of a system, a dual-layer iteration power allocation algorithm is proposed to eliminate the complex interference. The outer-layer iteration is to solve the non-linear fractional objective function based on Dinkelbach, and the inner-layer iteration is to solve the non-convex optimization problem based on D.C. programming. Then, the non-convex and non-linear fractional objective function is transformed into a convex function to solve the optimal power allocation. In this approach, FD D2D users reuse the spectrum with downlink NOMA cellular users. Imperfect self-interference (SI) cancellation at the FD D2D users and the successive interference cancellation (SIC) at the strong NOMA user are considered in the system framework. The optimization problem is constructed to maximize the system’s energy efficiency with the constraints of successful SIC, QoS requirements, the maximum transmit power of BS and FD D2D users. Numerical results demonstrate that the proposed algorithm outperforms the traditional orthogonal multiple access (OMA) in terms of energy efficiency with a higher system sum rate.

Airfoil shape optimization using genetic algorithm coupled deep neural networks

To alleviate the computational burden associated with the computational fluid dynamics (CFD) simulation stage and improve aerodynamic optimization efficiency, this work develops an innovative procedure for airfoil shape optimization, which is implemented through coupling the genetic algorithm (GA) optimizer with the aerodynamic coefficients prediction network (ACPN) model. The ACPN is established using a fully connected neural network with the airfoil geometry as the input and aerodynamic coefficients as the output. The results show that the ACPN's mean prediction accuracy for the lift and drag coefficient is high up to about 99.02%. Moreover, the prediction time of each aerodynamic coefficient is within 5 ms, four orders of magnitude faster compared to the CFD solver (3 min). Taking advantage of the fast and accurate prediction, the proposed ACPN model replaces the expensive CFD simulations and couples with GA to force the airfoil shape change to maximize the lift–drag ratio under multiple constraints. In terms of time efficiency, optimized airfoils can be fast obtained within 25 s. Even considering an extra 50 h spent on data preparing and 20 s for model training, the overall calculation cost is reduced by a remarkable 62.1% compared to the GA-CFD optimization method (5.5 days). Furthermore, the GA-ACPN model improves the lift–drag ratio with and without constraint by 51.4% and 55.4% for NACA0012 airfoil, respectively, while 50.3% and 60.0% improvement achieved by the GA-CFD optimization method. These results indicate that the GA-ACPN optimization approach significantly enhances the optimization efficiency and has great potential to address varying constraint optimization problems.

A Methodology for Fault-tolerant Pareto-optimal Approximate Designs of FPGA-based Accelerators

Approximate Computing Techniques (ACTs) take advantage of resilience computing applications to trade off among output precision, area, power, and performance. ACTs can lead to significant gains at affordable costs when efficiently implemented on Field Programmable Gate Array– (FPGA) based accelerators. Although several novel ACTs works have been proposed for FPGA accelerators, their applicability to high-assurance systems has not been explored as much. ACTs are becoming necessary in many critical Edge computing systems, such as self-driving cars and Earth observation satellites, to increase computational efficiency. However, an important question comes to mind when targeting critical systems: Does ACT optimization negatively affect the reliability of the system and how can one find optimal design architectures that blend classic mitigation techniques like Triple Modular Redundancy with approximation- and precise-based arithmetic hardware units to achieve the best possible computational efficiency without compromising dependability? This work aims to solve this research problem by introducing a Design Space Exploration (DSE) methodology that employs ACTs in arithmetic units of the design and identifies Pareto-optimal microarchitectures that balance all relevant gains of ACTs, such as area, speed, power, failure rate, and precision, by inserting the correct amount of approximation in the design. In a nutshell, our DSE methodology has formulated the DSE with a Multi-Objective Optimization Problem (MOP). Each Pareto-optimal solution of our tool finds which arithmetic units of the design to implement with precise and approximate circuits and which units to selectively triplicate to remove single points of failure that compromise system reliability below acceptable thresholds. We also suggest another formulation of the DSE into a Single-Objective constraint Optimization Problem (ScOP) producing a single optimal point, and that the user may demand, as a less time-consuming alternative to the MOP if a complete Pareto-front is not needed. Our methodology generates fault-tolerant versions of the Pareto-optimal approximate designs (or simple optimized approximate designs if the ScOP choice is picked) by selectively applying mitigation techniques in a way that the overheads of redundant resources for fault-tolerance do not negate the gains of approximation in comparison to the fault-tolerant versions of the precise design. We evaluate our method on two FPGA-based accelerators: a JPEG encoder and an H.264/Advanced Video Coding decoder. Our experimental results show significant gains in area, frequency, and power consumption without compromising output quality and system reliability compared to classic solutions that replicate all or a part of the resources of the precise design to increase dependability metrics.

Automatic generation of dominance breaking nogoods for a class of constraint optimization problems

Constraint Optimization Problems (COPs) ask for an assignment of values to variables in order to optimize an objective subject to constraints that restrict the value combinations in the assignment. They are usually solved by the classical Branch and Bound (B&B) search algorithm. Dominance breaking is an important technique in B&B to prune assignments that are subordinate to others concerning the objective value and/or the satisfiability of constraints. In practice, the addition of constraints for dominance breaking can drastically speed up the B&B search for solving many COPs. However, identification of suboptimal assignments in COPs and derivation of useful constraints for dominance breaking are usually problem-specific and require sophisticated human insights on the problem structure.This paper proposes the first theoretical and practical framework for automatic generation of dominance breaking constraints for a class of COPs consisting of efficiently checkable objectives and constraints. In particular, the framework focuses on generating nogood constraints representing incompatible value assignments and formulates nogood generation as solving auxiliary constraint satisfaction problems. The proposed method can generate nogoods of varying strengths for dominance breaking by controlling the number of involved variables. Experimentation on various benchmarks demonstrates the effectiveness of the proposal in both efficiency and ease of use. The superior performance is also supported by a theoretical analysis to compare the relative strength of automatically generated nogoods with manually derived dominance breaking constraints in the literature.

Integrated single target sensing and multiuser communications based on zero-forcing beamforming

Integrated sensing and communication (ISAC) is a promising approach that utilizes a single waveform for both information transmission and target object acquisition. In this study, we focus on the challenges posed by inter-user and inter-ISAC-antenna interference, specifically the self-interference from ISAC transmit antennas to receive antennas. Further, the benefit of a multi-functional ISAC scheme is clarified by comparing it with single function-aware schemes: sensing- and communication-aware approaches. To address these issues, we investigate the application of a zero-forcing (ZF)-based beamforming and power allocation (PA) technique. A sum rate maximization problem with a target object sensing signal-to-noise (SNR) threshold is formulated and split into two subproblems for improved tractability. To tackle the first subproblem of precoder design, we develop a constraint-aware greedy algorithm that employs communication SNR ordering. Subsequently, we propose a feasibility test-based algorithm to solve the second subproblem of PA.Through an extensive performance evaluation, encompassing computational complexity, sum rate, and ISAC received SNR, we verify that relying solely on a single function-aware scheme is insufficient for accommodating ISAC's multifunctional capabilities. Additionally, we demonstrate that a naive switching strategy between two single function-aware schemes can result in inefficient communication performance. Considering both sensing and communication functions, we identify the need for an adaptive balance between sensing and communication capabilities in a multi-functional ISAC system. This can be achieved by dynamically adjusting the target object sensing SNR threshold within the optimization problem's constraint. The proposed ZF-based ISAC with the designed PA scheme offers an optimal strategy when the transmit power of the ISAC is low. Conversely, an equal PA to multiple users emerges as the best strategy when the transmit power of the ISAC is sufficiently high. In conclusion, our study highlights the significance of integrating sensing and communication functionalities in ISAC systems. Moreover, by providing insights into the interplay between target object sensing, beamforming, and power allocation, we contribute to the development of efficient and adaptable ISAC architectures.

NOMA-based energy efficient resource allocation in wireless energy harvesting sensor networks

Non Orthogonal Multiple Access (NOMA) and energy harvesting are two widely used concepts to improve the performance of Wireless Sensor Networks. In this paper, we consider a NOMA-based Wireless Energy Harvesting Sensor Network (WEHSN), in which all sensors harvest their energy from a Hybrid Access Point (HAP), and then, simultaneously transmit their information in NOMA-based protocol to the HAP, if their harvested energy are sufficient and they have data for transmission. We formulate the resource allocation as an optimization problem to maximize the energy efficiency, defined as system throughput per total energy consumption in the network, in which the total energy consumption is summation of energy consumption in downlink and uplink phases. The optimization problem constraints are the time scheduling parameters and the sensors’ transmission powers. To solve this maximization problem, we use the Dinkelbach method to convert it to the parametric form and then, we apply the Karush–Kuhn–Tucker (KKT) conditions in the parametric optimization problem to derive the closed form expression for optimization problem. We evaluate the performance of proposed scheme in relation to energy efficiency, throughput and total energy consumption, and we compare them with OFDMA and TDMA-based WEHSN method. The numerical results show that, NOMA-based WEHSN has better performance in compared with OFDMA and TDMA-based WEHSN. Also, the effect of different amount of data at the sensors has been considered in numerical result which shows that when the sensors always have data for transmission, the system has better performance in terms of energy efficiency and throughput but it has more total energy consumption in comparison to the situation that the sensors occasionally have data for transmission.

An Adaptive Ant Colony Algorithm Based on Local Information Entropy to Solve Distributed Constraint Optimization Problems

As a meta-heuristic algorithm, the ant colony algorithm has been successfully used to solve various combinatorial optimization problems. However, the existing algorithm that takes the power of ants to solve distributed constraint optimization problems (ACO_DCOP) is easy to fall into local optima. To deal with this issue, this paper presents an adaptive ant colony algorithm based on local information entropy to solve distributed constraint optimization problems, named LIEAD. In LIEAD, the local information entropy is introduced to help agents adaptively select the pheromone update strategy and value selection strategy, which improves the convergence speed and the quality of the solution. Moreover, a restart mechanism is designed to break the accumulation state of pheromone, which increases the population diversity and helps the algorithm jump out of the local optima. The extensive experimental results indicate that LIEAD can significantly outperform ACO_DCOP and is competitive with the state-of-the-art DCOPs algorithms.

Finding Good Partial Assignments during Restart-Based Branch and Bound Search

Restart-based Branch-and-Bound Search (BBS) is a standard algorithm for solving Constraint Optimization Problems (COPs). In this paper, we propose an approach to find good partial assignments to jumpstart search at each restart for general COPs, which are identified by comparing different best solutions found in different restart runs. We consider information extracted from historical solutions to evaluate the quality of the partial assignments. Thus the good partial assignments are dynamically updated as the current best solution evolves. Our approach makes restart-based BBS explore different promising sub-search-spaces to find high-quality solutions. Experiments on the MiniZinc benchmark suite show how our approach brings significant improvements to a black-box COP solver equipped with the state of the art search techniques. Our method finds better solutions and proves optimality for more instances.

Learning to Break Symmetries for Efficient Optimization in Answer Set Programming

The ability to efficiently solve hard combinatorial optimization problems is a key prerequisite to various applications of declarative programming paradigms. Symmetries in solution candidates pose a significant challenge to modern optimization algorithms since the enumeration of such candidates might substantially reduce their performance. This paper proposes a novel approach using Inductive Logic Programming (ILP) to lift symmetry-breaking constraints for optimization problems modeled in Answer Set Programming (ASP). Given an ASP encoding with optimization statements and a set of small representative instances, our method augments ground ASP programs with auxiliary normal rules enabling the identification of symmetries using existing tools, like SBASS. Then, the obtained symmetries are lifted to first-order constraints with ILP. We prove the correctness of our method and evaluate it on real-world optimization problems from the domain of automated configuration. Our experiments show significant improvements of optimization performance due to the learned first-order constraints.

NuWLS: Improving Local Search for (Weighted) Partial MaxSAT by New Weighting Techniques

Maximum Satisfiability (MaxSAT) is a prototypical constraint optimization problem, and its generalized version is the (Weighted) Partial MaxSAT problem, denoted as (W)PMS, which deals with hard and soft clauses. Considerable progress has been made on stochastic local search (SLS) algorithms for solving (W)PMS, which mainly focus on clause weighting techniques. In this work, we identify two issues of existing clause weighting techniques for (W)PMS, and propose two ideas correspondingly. First, we observe that the initial values of soft clause weights have a big effect on the performance of the SLS solver for solving (W)PMS, and propose a weight initialization method. Second, we propose a new clause weighting scheme that for the first time employs different conditions for updating hard and soft clause weights. Based on these two ideas, we develop a new SLS solver for (W)PMS named NuWLS. Through extensive experiments, NuWLS performs much better than existing SLS solvers on all 6 benchmarks from the incomplete tracks of MaxSAT Evaluations (MSEs) 2019, 2020, and 2021. In terms of the number of winning instances, NuWLS outperforms state-of-the-art SAT-based incomplete solvers on all the 6 benchmarks. More encouragingly, a hybrid solver that combines NuWLS and an SAT-based solver won all four categories in the incomplete track of the MaxSAT Evaluation 2022.

Separate but Equal: Equality in Belief Propagation for Single Cycle Graphs

Belief propagation is a widely used incomplete optimization algorithm, whose main theoretical properties hold only under the assumptions that beliefs are not equal. Nevertheless, there is much evidence that equality between beliefs does occur. A method to overcome belief equality by using unary function-nodes is assumed to resolve the problem. We focus on Min-sum, the belief propagation version for solving constraint optimization problems. We prove that on a single cycle graph, belief equality can be avoided only when the algorithm converges to the optimal solution. In any other case, the unary function methods will not prevent equality, rendering some existing results in need of reassessment. We differentiate between belief equality, which includes equal beliefs in a single message, and assignment equality, that prevents a coherent selection of assignments to variables. We show the necessary and satisfying conditions for both.

Constraint Optimization over Semirings

Interpretations of logical formulas over semirings (other than the Boolean semiring) have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula phi over n variable and a semiring (K,+, . ,0,1), find the maximum value over all possible interpretations of phi over K. This can be seen as a generalization of the well-known satisfiability problem (a propositional formula is satisfiable if and only if the maximum value over all interpretations/assignments over the Boolean semiring is 1). A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We first show that for general propositional formulas in negation normal form, optConfVal and optConf are in FP^NP. We then investigate optConf when the input formula phi is represented in the conjunctive normal form. For CNF formulae, we first derive an upper bound on the value of optConf as a function of the number of maximum satisfiable clauses. In particular, we show that if r is the maximum number of satisfiable clauses in a CNF formula with m clauses, then its optConf value is at most 1/4^(m-r). Building on this we establish that optConf for CNF formulae is hard for the complexity class FP^NP[log]. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings.