Showing papers on "Constraint satisfaction published in 2022"

Balancing Objective Optimization and Constraint Satisfaction in Constrained Evolutionary Multiobjective Optimization

Abstract: Both objective optimization and constraint satisfaction are crucial for solving constrained multiobjective optimization problems, but the existing evolutionary algorithms encounter difficulties in striking a good balance between them when tackling complex feasible regions. To address this issue, this article proposes a two-stage evolutionary algorithm, which adjusts the fitness evaluation strategies during the evolutionary process to adaptively balance objective optimization and constraint satisfaction. The proposed algorithm can switch between the two stages according to the status of the current population, enabling the population to cross the infeasible region and reach the feasible regions in one stage, and to spread along the feasible boundaries in the other stage. Experimental studies on four benchmark suites and three real-world applications demonstrate the superiority of the proposed algorithm over the state-of-the-art algorithms, especially on problems with complex feasible regions.

A System Level Approach to Tube-Based Model Predictive Control

Abstract: Robust tube-based model predictive control (MPC) methods address constraint satisfaction by leveraging an a priori determined tube controller in the prediction to tighten the constraints. This letter presents a system level tube-MPC (SLTMPC) method derived from the system level parameterization (SLP), which allows optimization over the tube controller online when solving the MPC problem, which can significantly reduce conservativeness. We derive the SLTMPC method by establishing an equivalence relation between a class of robust MPC methods and the SLP. Finally, we show that the SLTMPC formulation naturally arises from an extended SLP formulation and show its merits in a numerical example.

A tri-population based co-evolutionary framework for constrained multi-objective optimization problems

Abstract: Balancing between the optimization of objective functions and constraint satisfaction is essential to handle constrained multi-objective optimization problems (CMOPs). Recently, various methods have been presented to enhance the performance for the constrained multi-objective optimization evolutionary algorithms (CMOEAs). However, most of them encounter difficulties when dealing with the CMOPs with complex feasible regions. To overcome this drawback, this paper proposes a tri-population based co-evolutionary framework (TriP): i) the first and second populations are evolved through a weak co-evolutionary relation for the original and unconstrained problems respectively to handle CMOPs with relatively simple constraints; and ii) the third population is evolved solely for the constraint relaxed problem with constraint relaxation technique. The cooperation of three populations preserve the advantages of weak co-evolution and constraint relaxation. Experiments on six benchmark CMOPs with 65 instances and diverse features are performed. Compared to 9 state-of-the-art CMOEAs, the proposed framework yields highly competitive performance and the best versatility. In addition, the effectiveness of the proposed framework on handling real-world CMOPs is also verified.

A System Level Approach to Tube-Based Model Predictive Control

Abstract: Robust tube-based model predictive control (MPC) methods address constraint satisfaction by leveraging an a priori determined tube controller in the prediction to tighten the constraints. This letter presents a system level tube-MPC (SLTMPC) method derived from the system level parameterization (SLP), which allows optimization over the tube controller online when solving the MPC problem, which can significantly reduce conservativeness. We derive the SLTMPC method by establishing an equivalence relation between a class of robust MPC methods and the SLP. Finally, we show that the SLTMPC formulation naturally arises from an extended SLP formulation and show its merits in a numerical example.

An invitation to the promise constraint satisfaction problem

Abstract: The study of the complexity of the constraint satisfaction problem (CSP), centred around the Feder-Vardi Dichotomy Conjecture, has been very prominent in the last two decades. After a long concerted effort and many partial results, the Dichotomy Conjecture has been proved in 2017 independently by Bulatov and Zhuk. At about the same time, a vast generalisation of CSP, called promise CSP, has started to gain prominence. In this survey, we explain the importance of promise CSP and highlight many new very interesting features that the study of promise CSP has brought to light. The complexity classification quest for the promise CSP is wide open, and we argue that, despite the promise CSP being more general, this quest is rather more accessible to a wide range of researchers than the dichotomy-led study of the CSP has been.

A Soft Constrained MPC Formulation Enabling Learning From Trajectories With Constraint Violations

Abstract: In practical model predictive control (MPC) implementations, constraints on the states are typically softened to ensure feasibility despite unmodeled disturbances. In this work, we propose a soft constrained MPC formulation supporting polytopic terminal sets in half-space and vertex representation, which significantly increases the feasible set while maintaining asymptotic stability in case of constraint violations. The proposed formulation allows for leveraging system trajectories that violate state constraints to iteratively improve the MPC controller’s performance. To this end, we apply convex optimization techniques to obtain a data-driven terminal cost and set, which result in a quadratic MPC problem.

The Complexity of General-Valued Constraint Satisfaction Problems Seen from the Other Side

Abstract: The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the $k$-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory.

Data-Driven Tuning for Chance-Constrained Optimization: Two Steps Towards Probabilistic Performance Guarantees

Abstract: Parameters involved in the formulation of optimization problems are often partially unknown or random. A popular way to mitigate the effect of uncertainty is using joint chance constraints, which guarantee constraint satisfaction with high probability but are challenging to solve. In this letter, we analyze an approach for joint chance constrained problems that involves iteratively tuning problem parameters. We first show that existing naive approaches to tuning can lead to solutions without feasibility guarantees. We then introduce a two-step approach, where a tuning-based solution generation step is followed by an a posteriori solution verification step. A main challenge of the two-step approach is to guarantee that the solution generated in the first step has a high probability of being verified as feasible in the second step. We therefore analyze how the relationship between the feasibility criteria used in each step impacts the probability of obtaining a feasible solution. We demonstrate our results in a numerical case study of the optimal power flow problem.

Hyperbolic Knowledge Transfer with Class Hierarchy for Few-Shot Learning

Abstract: Few-shot learning (FSL) aims to recognize a novel class with very few instances, which is a challenging task since it suffers from a data scarcity issue. One way to effectively alleviate this issue is introducing explicit knowledge summarized from human past experiences to achieve knowledge transfer for FSL. Based on this idea, in this paper, we introduce the explicit knowledge of class hierarchy (i.e., the hierarchy relations between classes) as FSL priors and propose a novel hyperbolic knowledge transfer framework for FSL, namely, HyperKT. Our insight is, in the hyperbolic space, the hierarchy relation between classes can be well preserved by resorting to the exponential growth characters of hyperbolic volume, so that better knowledge transfer can be achieved for FSL. Specifically, we first regard the class hierarchy as a tree-like structure. Then, 1) a hyperbolic representation learning module and a hyperbolic prototype inference module are employed to encode/infer each image and class prototype to the hyperbolic space, respectively; and 2) a novel hierarchical classification and relation reconstruction loss are carefully designed to learn the class hierarchy. Finally, the novel class prediction is performed in a nearest-prototype manner. Extensive experiments on three datasets show our method achieves superior performance over state-of-the-art methods, especially on 1-shot tasks.

Adaptively robust nonlinear model predictive control based on attack identification

Abstract: Abstract Robust model predictive control (MPC) is an essential tool for control systems under uncertainty as it allows for constraint satisfaction even if disturbances occur. When a system suffers malicious attacks, in contrast to parametric uncertainties or known systems faults, it is difficult to specify tight uncertainty ranges within which possible disturbances lie. In this case, very conservative solutions or even infeasible problems are obtained. To address this issue, we propose an adaptively robust MPC scheme that adjusts the uncertainty ranges to available knowledge about the attackers. To this end, we combine a recently proposed method identifying unknown attacks on nonlinear systems with a multi-stage approach to robust MPC. We illustrate the potential of the method in a numerical case study with a distributed nonlinear system.

Robust Output Feedback MPC for LPV Systems Using Interval Observers

Abstract: This article addresses the problem of robust output feedback model predictive control for discrete-time, constrained, linear parameter-varying systems subject to (bounded) state and measurement disturbances. The vector of scheduling parameters is assumed to be an unmeasurable signal taking values in a given compact set. The proposed controller incorporates an interval observer, that uses the available measurement to update the set-membership estimation of the states, and an interval predictor, used in the prediction step of the model predictive control (MPC) algorithm. The resulting MPC scheme offers guarantees on recursive feasibility, constraint satisfaction, and input-to-state stability in the terminal set. Furthermore, this novel algorithm shows low computation complexity and ease of implementation (similar to conventional MPC schemes).

A Soft Constrained MPC Formulation Enabling Learning From Trajectories With Constraint Violations

Abstract: In practical model predictive control (MPC) implementations, constraints on the states are typically softened to ensure feasibility despite unmodeled disturbances. In this work, we propose a soft constrained MPC formulation supporting polytopic terminal sets in half-space and vertex representation, which significantly increases the feasible set while maintaining asymptotic stability in case of constraint violations. The proposed formulation allows for leveraging system trajectories that violate state constraints to iteratively improve the MPC controller's performance. To this end, we apply convex optimization techniques to obtain a data-driven terminal cost and set, which result in a quadratic MPC problem.

A Hybrid Artificial Immune-Simulated Annealing Algorithm for Multiroute Job Shop Scheduling Problem With Continuous Limited Output Buffers

Abstract: In this article, we study the multiroute job shop scheduling problem with continuous-limited output buffers (MRJSP-CLOBs). In contrast to the standard job shop scheduling problem (JSP), continuous-limited output buffers render the commonly used graph-based approaches inapplicable, and the multiroute issue further increases computational complexity. To this end, we formulate MRJSP-CLOB as a mixed-integer linear program (MILP), which is typically NP-hard. Then, we extend the critical block in the JSP by utilizing the no-time-gap relationship and design a new neighborhood structure. Furthermore, we propose a hybrid artificial immune-simulated annealing algorithm (AIA-SA) by sharing iterations and integrating a random infeasible solution repairing algorithm with a new SA acceptance rule, which enables individuals to share information and increases the robustness of the corresponding SA parameters. Finally, the AIA-SA is compared with CPLEX and state-of-the-art algorithms on MRJSP-CLOB with different sizes. Experiments for large-sized instances demonstrate that our algorithm requires less than 3% computing time of the CPLEX, while being faster and more accurate than the other algorithms.

Discrete Mother Tree Optimization and Swarm Intelligence for Constraint Satisfaction Problems

Abstract: The Constraint Satisfaction Problem (CSP) is a powerful framework for a wide variety of combinatorial problems. The CSP is known to be NP-complete, and many algorithms have been developed to tackle this challenge in practice. These algorithms include the backtracking technique, improved with constraint propagation and variable ordering heuristics. Despite its success, backtracking still suffers from its exponential time cost, especially for large to solve problems. Metaheuristics, including local search and nature-inspired methods, can be an alternative that trades running time for the quality of the solution. Indeed, these techniques do not guarantee to return a complete solution, nor can they prove the inconsistency of the problem. They are, however time-efficient, thanks to their polynomial running time. In particular, nature-inspired techniques can be very effective if designed with a good exploitation/exploration balance during the search. To solve CSPs, we propose two discrete variants of two known nature-inspired algorithms. The first one is an adaptation of the Mother Tree Optimization (MTO). In contrast, the second is an extension of the Particle Swarm Optimization (PSO) with a new operator that we propose. Both variants rely on a heuristic that gathers information about constraints violations during the search. The latter will then be used to update candidate solutions, following a given topology for MTO, and position/velocity equations for PSO. To assess the performance of both methods, we conducted several comparative experiments, considering other known systematic methods and metaheuristics. The results demonstrate the effectiveness of both methods.

A collection of Constraint Programming models for the three-dimensional stable matching problem with cyclic preferences

Abstract: We introduce five constraint models for the 3-dimensional stable matching problem with cyclic preferences and study their relative performances under diverse configurations. While several constraint models have been proposed for variants of the two-dimensional stable matching problem, we are the first to present constraint models for a higher number of dimensions. We show for all five models how to capture two different stability notions, namely weak and strong stability. Additionally, we translate some well-known fairness notions (i.e. sex-equal, minimum regret, egalitarian) into 3-dimensional matchings, and present how to capture them in each model. Our tests cover dozens of problem sizes and four different instance generation methods. We explore two levels of commitment in our models: one where we have an individual variable for each agent (individual commitment), and another one where the determination of a variable involves pairing the three agents at once (group commitment). Our experiments show that the suitability of the commitment depends on the type of stability we are dealing with, and that the choice of the search heuristic can help improve performance. Our experiments not only brought light to the role that learning and restarts can play in solving this kind of problems, but also allowed us to discover that in some cases combining strong and weak stability leads to reduced runtimes for the latter.

Penalized Proximal Policy Optimization for Safe Reinforcement Learning

Abstract: Safe reinforcement learning aims to learn the optimal policy while satisfying safety constraints, which is essential in real-world applications. However, current algorithms still struggle for efficient policy updates with hard constraint satisfaction. In this paper, we propose Penalized Proximal Policy Optimization (P3O), which solves the cumbersome constrained policy iteration via a single minimization of an equivalent unconstrained problem. Specifically, P3O utilizes a simple yet effective penalty approach to eliminate cost constraints and removes the trust-region constraint by the clipped surrogate objective. We theoretically prove the exactness of the penalized method with a finite penalty factor and provide a worst-case analysis for approximate error when evaluated on sample trajectories. Moreover, we extend P3O to more challenging multi-constraint and multi-agent scenarios which are less studied in previous work. Extensive experiments show that P3O outperforms state-of-the-art algorithms with respect to both reward improvement and constraint satisfaction on a set of constrained locomotive tasks.

SDPs and Robust Satisfiability of Promise CSP

Abstract: For a constraint satisfaction problem (CSP), a robust satisfaction algorithm is one that outputs an assignment satisfying most of the constraints on instances that are near-satisfiable. It is known that the CSPs that admit efficient robust satisfaction algorithms are precisely those of bounded width, i.e., CSPs whose satisfiability can be checked by a simple local consistency algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact satisfiability of a bounded width CSP can be checked by combinatorial algorithms, the robust algorithm is based on rounding a canonical Semi Definite Programming(SDP) relaxation. In this work, we initiate the study of robust satisfaction algorithms for promise CSPs, which are a vast generalization of CSPs that have received much attention recently. The motivation is to extend the theory beyond CSPs, as well as to better understand the power of SDPs. We present robust SDP rounding algorithms under some general conditions, namely the existence of majority or alternating threshold polymorphisms. On the hardness front, we prove that the lack of such polymorphisms makes the PCSP hard for all pairs of symmetric Boolean predicates. Our method involves a novel method to argue SDP gaps via the absence of certain colorings of the sphere, with connections to sphere Ramsey theory. We conjecture that PCSPs with robust satisfaction algorithms are precisely those for which the feasibility of the canonical SDP implies (exact) satisfiability. We also give a precise algebraic condition, known as a minion characterization, of which PCSPs have the latter property.

A residual-based message passing algorithm for constraint satisfaction problems

Abstract: Message passing algorithms, whose iterative nature captures complicated interactions among interconnected variables in complex systems and extracts information from the fixed point of iterated messages, provide a powerful toolkit in tackling hard computational tasks in optimization, inference, and learning problems. In the context of constraint satisfaction problems (CSPs), when a control parameter (such as constraint density) is tuned, multiple threshold phenomena emerge, signaling fundamental structural transitions in their solution space. Finding solutions around these transition points is exceedingly challenging for algorithm design, where message passing algorithms suffer from a large message fluctuation far from convergence. Here we introduce a residual-based updating step into message passing algorithms, in which messages with large variation between consecutive steps are given high priority in the updating process. For the specific example of model RB (revised B), a typical prototype of random CSPs with growing domains, we show that our algorithm improves the convergence of message updating and increases the success probability in finding solutions around the satisfiability threshold with a low computational cost. Our approach to message passing algorithms should be of value for exploring their power in developing algorithms to find ground-state solutions and understand the detailed structure of solution space of hard optimization problems.

Recursively Feasible Stochastic Predictive Control Using an Interpolating Initial State Constraint

Abstract: We present a stochastic model predictive control (SMPC) framework for linear systems subject to possibly unbounded disturbances. State of the art SMPC approaches with closed-loop chance constraint satisfaction recursively initialize the nominal state based on the previously predicted nominal state or possibly the measured state under some case distinction. We improve these initialization strategies by allowing for a continuous optimization over the nominal initial state in an interpolation of these two extremes. The resulting SMPC scheme can be implemented as one standard quadratic program and is more flexible compared to state-of-the-art initialization strategies. As the main technical contribution, we show that the proposed SMPC framework also ensures closed-loop satisfaction of chance constraints and suitable performance bounds.

Set-Theoretic Failure Mode Reconfiguration for Stuck Actuators

Abstract: This letter proposes a set-theoretic Failure Mode and Effect Management (FMEM) strategy that handles stuck/jammed actuators and enforces pointwise-in-time state and control constraints. The approach exploits nesting between constraint admissible and recoverable sets to ensure the existence of a recovery sequence. A reference governor is applied to track reference commands while imposing constraint satisfaction using the remaining working actuators. Numerical results of an aircraft longitudinal flight application are reported.

Beyond PCSP(1-in-3,NAE)

Abstract: The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs. 1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SICOMP'21] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (1-in-3,NAE) is tractable. We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (t-in-k,NAE) by either adding tuples to t-in-k or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek, Guruswami, Wrochna, and \v{Z}ivn\'y [SICOMP'20]. For the latter, we classify all templates as either tractable or NP-hard.

Scalable tube model predictive control of uncertain linear systems using ellipsoidal sets

Abstract: This work proposes a novel robust model predictive control (MPC) algorithm for linear systems affected by dynamic model uncertainty and exogenous disturbances. The uncertainty is modeled using a linear fractional perturbation structure with a time-varying perturbation matrix, enabling the algorithm to be applied to a large model class. The MPC controller constructs a state tube as a sequence of parameterized ellipsoidal sets to bound the state trajectories of the system. The proposed approach results in a semidefinite program to be solved online, whose size scales linearly with the order of the system. The design of the state tube is formulated as an offline optimization problem, which offers flexibility to impose desirable features such as robust invariance on the terminal set. This contrasts with most existing tube MPC strategies using polytopic sets in the state tube, which are difficult to design and whose complexity grows combinatorially with the system order. The algorithm guarantees constraint satisfaction, recursive feasibility, and stability of the closed loop. The advantages of the algorithm are demonstrated using two simulation studies.