Complementarity Conditions Research Articles

A new neural network based on smooth function for SOCCVI problems

A new neural network is proposed to solve the second-order cone constrained variational inequality (SOCCVI) problems. Instead of the smoothed Fishcer-Burmeister function, a smooth regularized Chen-Harker-Kanzow-Smale (CHKS) function is used to handle relevant complementarity conditions. By using a neural network approach based on the CHKS function, the KKT conditions corresponding to the SOCCVI are solved. Some stability properties of the neural network can be verified by the Lyapunov method. When the parameters of the neural network are different, the achieved convergence speed will also vary. Further by controlling the corresponding parameters, the neural network can achieve a faster convergence speed than a classical model. Numerical simulations are applied to examine the computing capability of the neural network as well as the influence of parameters on it.

Linear-Quadratic Mean-Field-Type Difference Games With Coupled Affine Inequality Constraints

In this letter, we study a class of linear-quadratic mean-field-type difference games with coupled affine inequality constraints. We show that the mean-filed-type equilibrium can be characterized by the existence of a multiplier process which satisfies some implicit complementarity conditions. Further, we show that the equilibrium strategies can be computed by reformulating these conditions as a single large-scale linear complementarity problem. We illustrate our results with an energy storage problem arising in the management of microgrids.

On the Efficient Implementation of the Matrix Exponentiated Gradient Algorithm for Low-Rank Matrix Optimization

Convex optimization over the spectrahedron, that is, the set of all real n × n positive semidefinite matrices with unit trace, has important applications in machine learning, signal processing, and statistics, mainly as a convex relaxation for optimization problems with low-rank matrices. It is also one of the most prominent examples in the theory of first order methods for convex optimization in which non-Euclidean methods can be significantly preferable to their Euclidean counterparts. In particular, the desirable choice is the matrix exponentiated gradient (MEG) method, which is based on the Bregman distance induced by the (negative) von Neumann entropy. Unfortunately, implementing MEG requires a full singular value decomposition (SVD) computation on each iteration, which is not scalable to high-dimensional problems. In this work, we propose efficient implementations of MEG, with both deterministic and stochastic gradients, which are tailored for optimization with low-rank matrices, and only use a single low-rank SVD computation on each iteration. We also provide efficiently computable certificates for the correct convergence of our methods. Mainly, we prove that, under a strict complementarity condition, the suggested methods converge from a warm-start initialization with similar rates to their full SVD–based counterparts. Finally, we bring empirical experiments that both support our theoretical findings and demonstrate the practical appeal of our methods.Funding: This work was supported by the Israel Science Foundation [Grant 1108/18].

Dynamic Complementarity Conditions and Whole-Body Trajectory Optimization for Humanoid Robot Locomotion

This article presents a planner to generate walking trajectories by using the centroidal dynamics and the full kinematics of a humanoid robot. The interaction between the robot and the walking surface is modeled explicitly via new conditions, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic complementarity conditions</i> . The approach does not require a predefined contact sequence and generates the footsteps automatically. We characterize the robot control objective via a set of tasks, and we address it by solving an optimal control problem. We show that it is possible to achieve walking motions automatically by specifying a minimal set of references, such as a constant desired center of mass velocity and a reference point on the ground. Furthermore, we analyze how the contact modeling choices affect the computational time. We validate the approach by generating and testing walking trajectories for the humanoid robot iCub.

A Cooperative Energy Transaction Model for VPP Integrated Renewable Energy Hubs in Deregulated Electricity Markets

This article proposes a cooperative energy transaction model for a virtual power plant (VPP) developed as a bilevel optimization program based on the Stackelberg game in the deregulated electricity markets. The VPP integrates multiple renewable energy hubs consisting of electric vehicle charging stations with photovoltaic and battery energy storage systems. The proposed model derives co-optimized strategic decisions for the VPP operator in the day-ahead (joint active power, reserve, and reactive power) markets while considering the balancing market operation. The proposed model is reformulated as a mathematical program with equilibrium constraints (MPECs) using the Karush–Kuhn–Tucker conditions and the strong duality theorem. For an efficient solution, the MPEC is approximated as a mixed-integer linear program by applying the Fortuny-Amat transform on the complementarity and slackness conditions and the binary expansion on the bilinear terms. The uncertainties of the renewables, loads, and prices are modeled using stochastic scenarios. Finally, comparative case studies of a VPP in an IEEE 33 node active distribution network show that the proposed model improves the profit by strategic cooperative energy transactions.

A Penalty Branch-and-Bound Method for Mixed Binary Linear Complementarity Problems

Linear complementarity problems (LCPs) are an important modeling tool for many practically relevant situations and also have many important applications in mathematics itself. Although the continuous version of the problem is extremely well-studied, much less is known about mixed-integer LCPs (MILCPs) in which some variables have to be integer-valued in a solution. In particular, almost no tailored algorithms are known besides reformulations of the problem that allow us to apply general purpose mixed integer linear programming solvers. In this paper, we present, theoretically analyze, enhance, and test a novel branch-and-bound method for MILCPs. The main property of this method is that we do not “branch” on constraints as usual but by adding suitably chosen penalty terms to the objective function. By doing so, we can either provably compute an MILCP solution if one exists or compute an approximate solution that minimizes an infeasibility measure combining integrality and complementarity conditions. We enhance the method by MILCP-tailored valid inequalities, node selection strategies, branching rules, and warm-starting techniques. The resulting algorithm is shown to clearly outperform two benchmark approaches from the literature. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms – Discrete. Funding: M. De Santis acknowledges support within the project RM120172A2970290, which has received funding from Sapienza, University of Rome. M. Schmidt thanks the Deutsche Forschungsgemeinschaft (DFG) for its support within project A05 and B08 in the “SFB TRR 154 Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks.” L. Winkel is supported by the DFG within the Research Training Group 2126: “Algorithmic Optimization.” Supplemental Material: The online supplementary material is available at https://doi.org/10.1287/ijoc.2022.1216 .

Homogenization for polynomial optimization with unbounded sets

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work.

Optimization and Performance Analysis of Francis Turbine Runner Based on Super-Transfer Approximate Method under Multi-Energy Complementary Conditions

Hydropower unit is the compensation power generation of the energy regulating unit in wind–solar–water multi-energy complementary systems that often require the turbine to operate in a partial working condition area, thereby resulting in problems of low hydraulic efficiency and severe vibration during operation. A multi-objective and multi-condition optimization design method for Francis turbine runner based on the super-transfer approximation approach was proposed in this study. The proposed method aims to improve the hydraulic performance of the turbine, enhance and suppress the vibration of the turbine, and expand the operation range of the turbine on the basis of the actual situation given that Francis turbine frequently operates in low- and ultralow-load areas under the condition of multi-energy complementarity and continuous adjustment of operating conditions. Different operating conditions from low load to full load were selected as performance evaluation conditions. The super-transfer approximation method was used to select the weight co-efficient of water turbine operating conditions, and a multi-objective optimization function with the efficiency and cavitation performance of the water turbine as optimization objectives was constructed to ensure that the optimized water turbine can achieve the optimal performance in the full working condition range. Results showed that the pressure distribution on the blade surface of the optimized runner was uniform and the working ability was enhanced under the condition of ensuring the performance stability of optimal and rated conditions when the original runner was optimized. The hydraulic efficiency of the turbine under the low-load conditions OP1 and OP2 increased by 4.61 and 3.17%, respectively. Hence, the optimized runner is suitable for hydraulic turbines under multi-energy complementary conditions. The results of this study can provide a reference for the optimal design and operation of the turbine runner.

Parallel solution of optimal control problems using the graphics processing unit

AbstractThis paper presents an indirect method for solving optimal control problems (OCPs) using graphics processing unit (GPU). The OCPs considered here include control variable inequality constraints (CVICs), state variable inequality constraints (SVICs), and parameters. The necessary conditions of the minimum for the OCPs are written as a boundary value problem with index‐1 differential algebraic equations (BVP‐DAEs). The complementarity conditions associated with those inequality constraints are approximated using Kanzow's smoothed Fisher‐Burmeister formula. The numerical solution for solving the BVP‐DAEs is based on the multiple shooting technique and the DAEs are solved using a single step linearly implicit Runge–Kutta (Rosenbrock–Wanner, ROW) method. A Newton's continuation method is performed to solve the BVP‐DAEs system and the descent direction is found by solving a sparse bordered almost block diagonal (BABD) linear system with a structured orthogonal factorization algorithm. Parallel computing techniques are used to accelerate the code which is implemented using Python and CUDA on GPU. Numerical examples are presented to illustrate the efficiency of the code. The GPU based parallel implementation is shown to be significantly faster than the implementation using central processing unit (CPU) alone.

Investigating How the Equilibria of the Electricity Market are Affected by Modeling the Strategic Behavior of Consumers

The paper proposes a formulation for a generalized Nash equilibrium model which incorporates the strategic biddings of some consumers capable of providing reserve and balancing services in the day-ahead and balancing markets, respectively. The strategic bidding in the electricity market is modelled using a bilevel optimization programming, where the electricity cost of consumers is minimized at the Upper Level (UL), and the energy and reserve costs in the market clearing process are co-optimized at the lower level (LL). By means of the strong duality theorem, the original bilevel model is transformed into an equivalent single-level Mathematical Problem with Equilibrium Constraints (MPEC). The joint problem of all MPECs, one per consumer, constitutes an Equilibrium Problem with Equilibrium Constraints (EPEC). The resulting EPEC is finally formulated as an auxiliary Mixed-Integer Linear Programming (MILP) problem. To this end, an exact linearization technique and Fortuny-Amat transformation are adopted to substitute the nonlinear terms and the complementarity conditions. In addition, a parametrization technique is used to replace the dual variable associated with the strong duality equation which appears in the EPEC problem. The diagonalization method is also adopted in this part as an ex-post analysis to verify the obtained solutions of the resulted MILP. Finally, a 3-bus illustrative example and the IEEE RTS 24-Bus System and 118-Bus System are considered to investigate the performance of the proposed approach.

FELINE: Finite element solver for hydrodynamic lubrication problems using the inexact Newton method

In this work we present FELINE, a C++ solver of the Reynolds equation for treating hydrodynamic lubrication problems. To correctly describe cavitation regions, FELINE implements the inexact Newton iteration (INE) algorithm within a finite element method (FEM) framework. The solver was tested and validated against known cases in literature and industrially relevant cases of dimpled textures. Furthermore, we provide a benchmark for a complex dimpled texture case to evaluate the performance and robustness of the implementation. FELINE performs very fast when compared with existing implementations and shows a great degree of stability, while providing physically correct solutions thanks to the INE algorithm. Program summaryProgram title: FELINECPC Library link to program files:https://doi.org/10.17632/srkpysb727.1Licensing provisions: Creative Commons by 4.0Programming language: C++Nature of problem: A fast Reynolds equation solver robust enough to correctly solve the linear complementarity problems that arise from proper treatment of the cavitation boundaries.Solution method: An inexact Newton iteration method implemented within a finite element method framework that properly ensures the complementarity conditions are satisfied by introducing a forcing term.

Shear-induced migration in concentrated suspensions: Particle mass conservation, contact pressure and jamming

A two-velocity fluid model is presented to describe particle migration in mono-disperse suspensions of neutrally buoyant particles. In contrast to previous migration models, the proposed formulation allows us to impose explicit boundary conditions on particle velocity, and thereby to satisfy strict mass conservation for the particle phase. In addition, the upper bound on particle volume fraction (jamming limit) is strictly enforced through a non-smooth complementarity condition and the introduction of a particle jamming pressure. The model is applied to an axisymmetric Poiseuille flow and solved using a finite-element method. For that purpose, a specific, fully implicit algorithm based on non-smooth optimisation tools is developed and validated. Preliminary comparisons with experimental data from the literature show promising agreement. In particular, the model properly captures the formation of an inner plug region, in which the suspension is saturated and jammed. • New particle migration model for suspension. • Strict mass conservation for each phase. • Two-velocity description. • Jamming via non-smooth complementarity condition.

Co-optimization of Energy and Reserve With Incentives to Wind Generation

Transitioning from fossil fuel dominated power systems to high penetrations of intermittent renewable generation is affecting classical electricity market designs. Here, a method is proposed to model and assess the absence of co-optimization of energy and reserve that prevails in the European system. Co-optimization is formulated through an optimization problem (COM), and its absence as an equilibrium problem (EQM) built on Karush-Kuhn-Tucker conditions of agents’ optimization and market clearing equations. EQM cannot be reformulated as a single optimization problem. Market distortions are identified by comparing the complementarity conditions of both models. These are then discussed on system with Feed in Premium to wind generation. Parameters in the models allow to represent different market configurations regarding: available wind generation, Feed in Premium to wind, generators’ risk aversion, and required reserve from wind generation.

Definition of industrial enterprises development conditions and new approaches to resources using

The article presents an analysis of the conditions required to provide the industrial sustainable development in the digital transformations. A study has been carried out that determines the complementary assets role in modelling the enterprises development management processes operating in the context of ecosystem interactions in order to determine the properties of complementary interactions that contribute to the further value formation as a result of controlled changes in interactions by activity type or by product type. It is proposed to establish interactions by economic activity types presented in the Russian Classification of Economic Activities, which will allow us to determine the maximum possible profitability as a condition of complementarity from the main activity type and additional incoming types. The complementarity signs, which make it possible to achieve the required development and sustainability level in the ecosystem, have been determined.

Oxy-fuel co-combustion dynamics of phytoremediation biomass and textile dyeing sludge: Gas-to-ash pollution abatement

The environmental pressures of major wastes in the circular economies can be abated leveraging the complementarity and optimal conditions of their co-combustion. The oxy-fuel co-combustion of phytoremediation biomass of Sedum alfredii Hance (SAH) and textile dyeing sludge (TDS) may be a promising choice for sustainable CO2 capture and a waste-to-energy conversion. This study characterized and quantified their co-combustion performances, kinetics, and interactions as a function of blend ratio, atmosphere type, and temperature. With a focus on the characteristic elements of SAH (Ca, K, Zn, and Cd) and TDS (Al and S), changes in the mineral phases and ash melting and slagging trends of K2O-Al2O3-SiO2 and CaO-Al2O3-SiO2 systems were quantified. The Zn and Cd residual rates of the co-combustion of 75% SAH and 25% TDS rose by 58.52% and 5.93%, respectively, in the oxy-fuel atmosphere at the 30% oxygen concentration, relative to the mono-combustion of SAH in the air atmosphere. The co-combustion in the oxy-fuel atmosphere at the 20% oxygen concentration delayed the release peaks of SO2, C2S, and H2S, while the Ca-rich SAH captured S in TDS through the formation of CaSO4. Our findings provide new and practical insights into the oxy-fuel co-combustion toward the enhanced co-circularity.

On linear problems with complementarity constraints

A mathematical program with complementarity constraints is an optimization problem with equality/inequality constraints in which a complementarity type constraint is considered in addition. This complementarity condition modifies the feasible region so as to remove many of those properties that are usually important to obtain the standard optimality conditions, e.g., convexity and constraint qualifications. In this paper, in the linear case, we introduce a decomposition method of the given problem in a sequence of parameterized problems, that aim to force complementarity. Once we obtain a feasible solution, by means of duality results, we are able to eliminate a set of parameterized problems which are not worthwhile to be considered. Furthermore, we provide some bounds for the optimal value of the objective function and we present an application of the proposed technique in a non trivial example and some numerical experiments.

Genetic diagnosis of ADPKD

Sequencing techniques represent the gold standard in molecular diagnosis ofautosomal dominant polycystic kidney disease (ADPKD), being in this case able to accurately and concretely highlight mutations at the genome level. The PKD1 gene mutation detection technique using the Real Time PCR technique and LNA hydrolysis probes exploits the specific behavior of hybridization LNA probes under the conditions of maximum complementarity with the target sequence and consequently the theoretical possibility of designing two such probes, one for the wild genotype (WT) and the other to identify the point mutation (MUT). The technique of identifying the PKD1 gene mutation using the Real Time PCR technique and FRET type hydrolysis probes consists of a derivation of the Real Time PCR technique, with the detection of amplicons by means of fluorescence with hybridization probes. The technique of identifying the PKD1 gene mutation using the Real Time PCR technique coupled with HRM (High Resolution Melting) involves one of the variants of the Real Time PCR technique, namely the interpretation of the results according to the melting temperature.

An open-source unconstrained stress updating algorithm for the modified Cam-clay model

This paper presents an unconstrained stress updating algorithm for a critical state plastic model of clay soil, where the loading/unloading estimations and the consideration of the stress behaviour transition from elasticity to plasticity can be bypassed by using the Fischer–Burmeister smoothing function instead of the Kuhn–Tucker complementarity conditions. A smoothing tangent operator consistent with the unconstrained stress updating strategy is derived from preserving the quadratic convergence speed for the global solution. Specifically, the relationship and difference between the consistent and continuum tangent operators are analysed from the perspectives of algebra and geometry. In addition, the nonlinear constitutive equations obtained by the backward Euler integration scheme are solved by the double dogleg trust region method (improved by non-monotonic technology), where a larger strain increment than that of the newton method is allowed for the stress updating. Then, the modified Cam-clay model for soil is implemented in ABAQUS/Standard by the proposed algorithm. The results of numerical examples show that the algorithm has significant advantages in terms of computation efficiency and robustness in contrast to the ABAQUS/Standard default integration algorithm, especially for the condition of large load increments and cyclic loadings. More than twice the computational efficiency of the ABAQUS/Standard default integration algorithm can be observed in the representative numerical examples. The source code of the proposed algorithm is freely available at https://github.com/zhouxin615/NMTR_Method.

Power and Operations. Simondon and the Imaginaries of the Nuclear Industry

In this paper, I use a theoretical framework inspired by Simondon to analyze the closed fuel cycle strategy implemented by the French nuclear industry in the 1970's. I confront the technocratic conception of technical ensembles, which sees them as the instantiation of a power over nature, with their technological understanding as systems of operations, i.e., points of mediation between technical invention and the natural environment. I argue that the closed fuel cycle strategy can be understood as relying on an imaginary ecology. I propose here a form of critical epistemology, which I compare with Jasanoff's theory of sociotechnical imaginaries, leading to a sociopolitical comprehension of the social efficiency and motives of such a representation. Finally, I question the complementarity conditions between those two frameworks, one normative and the other explanatory.

EQUIVALENCE OF THE FREDHOLM SOLVABILITY CONDITION FOR THE NEUMANN PROBLEM TO THE COMPLEMENTARITY CONDITION

The methods of complex analysis constitute the classical direction in the study of elliptic equations and mixed-type equations on the plane and fundamental results have now been obtained. In the early 60s of the last century, a new theoretical-functional approach was developed for elliptic equations and systems based on the use of functions analytic by Douglis. In the works of A.P. Soldatov and Yeh, it turned out that in the theory of elliptic equations and systems, Douglis analytic functions play an important role. These functions are solutions of a first-order elliptic system generalizing the classical Cauchy-Riemann system. In this paper, the Fredholm solvability of the generalized Neumann problem for a high-order elliptic equation on a plane is investigated. The equivalence of the solvability condition of the generalized Neumann problem with the complementarity condition (Shapiro-Lopatinsky condition) is proved. The formula for the index of the specified problem in the class of functions under study is calculated.