Parametric Quadratic Programming Research Articles

An Improved Dynamic Contact Model for Mass–Spring and Finite Element Systems Based on Parametric Quadratic Programming Method

AN IMPROVED DYNAMIC CONTACT MODEL FOR MASS–SPRING AND FINITE ELEMENT SYSTEMS IS PROPOSED IN THIS PAPER. THE PROPOSED MODEL AVOIDS THE NUMERICAL TROUBLES OF SPURIOUS HIGH-FREQUENCY OSCILLATIONS FOR MASS–SPRING AND FINITE ELEMENT SYSTEMS IN DYNAMIC CONTACT PROBLEMS BY USING THE PARAMETRIC QUADRATIC PROGRAMMING TECHNIQUE. THE ITERATIVE PROCESS FOR DETERMINATION OF CONTACT STATES ARE NOT REQUIRED FOR EACH TIME STEP IN THE PROPOSED METHOD, AS THE CONTACT STATES ARE TRANSFORMED INTO THE BASE EXCHANGES IN THE SOLUTION OF A STANDARD QUADRATIC PROGRAMMING PROBLEM. THE PROPOSED METHODOLOGY IMPROVES STABILITY AND HAS GOOD CONVERGENCE BEHAVIOR FOR DYNAMIC CONTACT PROBLEMS. NUMERICAL RESULTS DEMONSTRATE THE VALIDITY OF THE PROPOSED METHOD.

On solving quadratically constrained quadratic programming problem with one non-convex constraint

In this paper we consider a quadratically constrained quadratic programming problem with convex objective function and many constraints in which only one of them is non-convex. This problem is transformed to a parametric quadratic programming problem without any non-convex constraint and then by solving the parametric problem via an iterative scheme and updating the parameter in each iteration, the solution of the problem is achieved. The convergence of the proposed method is investigated. Numerical examples are given to show the applicability of the new method.

Fully Decentralized Optimal Power Flow of Multi-Area Interconnected Power Systems Based on Distributed Interior Point Method

Traditionally, the optimal power flow (OPF) problem is solved in a centralized manner. However, with continuous expansion of the scale of multi-area interconnected power systems, realistic applications of the centralized method face additional challenges. In this paper, we propose a fully decentralized OPF algorithm for multi-area interconnected power systems based on the distributed interior point method, where solving the regional correction equation was converted into solving a parametric quadratic programming problem during each Newton-Raphson iteration. This is a novel approach to tackle the decentralized OPF problems. We analyzed the convergence property of this algorithm. The proposed approach is fully decentralized without central coordinator and parameter tuning, and is robust to network partitioning. Furthermore, this decentralized algorithm enjoys the same convergence performance and accuracy as the centralized interior point method. Results on a 3-bus test system, 4 IEEE test systems, and a real 4-area 6056-bus interconnected system show the benefits of the proposed method.

A Solution Path Algorithm for General Parametric Quadratic Programming Problem.

Parameter in learning problems (usually arising from the tradeoff between training error minimization and regularization) is often tuned by cross validation (CV). A solution path provides a compact representation of all optimal solutions, which can be used to determine the parameter with the global minimum CV error, without solving original optimization problems multiple times based on grid search. However, existing solution path algorithms do not provide a unified implementation to various learning problems. In this paper, we first introduce a general parametric quadratic programming (PQP) problem that can be instantiated to an extensive number of learning problems. Then, we propose a generalized solution path (GSP) for the general PQP problem. Particularly, we use the $QR$ decomposition to handle singularities in GSP. Finally, we analyze the finite convergence and the time complexity of GSP. Our experimental results on a variety of data sets not only confirm the identicality between GSP and several existing solution path algorithms but also show the superiority of our GSP over the existing solution path algorithms on both generalization and robustness. Finally, we provide a practical guild of using the GSP to solve two important learning problems, i.e., generalized error path and Ivanov SVM.

2-Phase NSGA II: An Optimized Reward and Risk Measurements Algorithm in Portfolio Optimization

Portfolio optimization is a serious challenge for financial engineering and has pulled down special attention among investors. It has two objectives: to maximize the reward that is calculated by expected return and to minimize the risk. Variance has been considered as a risk measure. There are many constraints in the world that ultimately lead to a non–convex search space such as cardinality constraint. In conclusion, parametric quadratic programming could not be applied and it seems essential to apply multi-objective evolutionary algorithm (MOEA). In this paper, a new efficient multi-objective portfolio optimization algorithm called 2-phase NSGA II algorithm is developed and the results of this algorithm are compared with the NSGA II algorithm. It was found that 2-phase NSGA II significantly outperformed NSGA II algorithm.

Effects of fuzzy data on decision making in a competitive supply chain

The essence of data plays a critical role in decision making in supply chain management (SCM). When data embedded in supply chains (SCs) are fuzzy, the associated equilibrium and performance measures also become fuzzy. This paper investigates the effects of fuzzy data on decision making in a two-echelon SC with a supplier and duopolistic retailers playing a Stackelberg strategic game in both intra- and inter-echelons. In contrast to existing approaches, this paper devises an analysis method to provide a likely interval of the fuzzy maximal profit with a known possibility level (degree of certainty) rather than a singleton (crisp value). The idea is based on the extension principle to reformulate the two-level optimization problem as a pair of parametric quadratic programs in order to calculate the lower and upper bounds of the leader’s fuzzy maximal profit at each possibility level of the obtained information. The analytic results indicate that the higher the degree of uncertainty, the smaller (larger) the lower (upper) bound of the maximum profit of each SC member. Moreover, the main results obtained from eight scenarios show that when the degree of demand diversity between the two retailers is significantly high, the Stackelberg leader is most likely to obtain lower profit and the marginal contribution of the primary demand to the total profit of the duopolistic retailers will exceed that of the powerful supplier’s maximum profit.

On analyzing and detecting multiple optima of portfolio optimization

Portfolio selection is widely recognized as the birth-place of modern finance; portfolio optimization has become a developed tool for portfolio selection by the endeavor of generations of scholars. Multiple optima are an important aspect of optimization. Unfortunately, there is little research for multiple optima of portfolio optimization. We present examples for the multiple optima, emphasize the risk of overlooking the multiple optima by (ordinary) quadratic programming, and report the software failure by parametric quadratic programming. Moreover, we study multiple optima of multiple-objective portfolio selection and prove the nonexistence of the multiple optima of an extension of the model of Merton. This paper can be a step-stone of studying the multiple optima.

Value of information in portfolio selection, with a Taiwan stock market application illustration

Despite many proposed alternatives, the predominant model in portfolio selection is still mean–variance. However, the main weakness of the mean–variance model is in the specification of the expected returns of the individual securities involved. If this process is not accurate, the allocations of capital to the different securities will in almost all certainty be incorrect. If, however, this process can be made accurate, then correct allocations can be made, and the additional expected return following from this is the value of information. This paper thus proposes a methodology to calculate the value of information. A related idea of a level of disappointment is also shown. How value of information calculations can be important in helping a mutual fund settle on how much to set aside for research is discussed in reference to a Taiwan Stock Exchange illustrative application in which the value of information appears to be substantial. Heavy use is made of parametric quadratic programming to keep computation times down for the methodology.

Interval quadratic programming for day-ahead dispatch of uncertain predicted demand

In this paper, we propose an interval quadratic programming method for the day-ahead scheduling of power generation and battery charge cycles, where the prediction uncertainty of power consumption and photovoltaic power generation is described as a parameter vector lying in an interval box. The interval quadratic programming is formulated as the problem of finding the tightest box, i.e., interval hull, that encloses the image of a function of the minimizer in parametric quadratic programming. To solve this problem in a computationally efficient manner, we take a novel approach based on a monotonicity analysis of the minimizer in the parametric quadratic programming. In particular, giving a tractable parameterization of the minimizer on the basis of the Karush–Kuhn–Tucker condition, we show that the monotonicity analysis with respect to the parameter vector can be relaxed to the sign pattern analysis of an oblique projection matrix. The monotonicity of the minimizer is found to be essential in the day-ahead dispatch problem, where uncertain predicted demand, described by a parameter vector, is dispatched to power generation and battery charge cycles while the economic cost is minimized. Finally, we verify the efficiency of the proposed method numerically, using experimental and predicted data for power consumption and photovoltaic power generation.

Analytical Approach in Computing Nash Equilibrium for Oligopolistic Competition of Transmission-Constrained GENCOs

This paper proposes an analytical solution for bidding strategy problem of transmission-constrained generating companies (GENCOs) in an hour-ahead electricity market. Here, for modeling a bidding strategy problem under a supply function equilibrium model, a bilevel programming (BLP) method is used. In the upper level, the profit of an intended GENCO is maximized, and in the lower level, the independent system operator clears the market via solving a bid-based security-constrained economic dispatch (SCED), in which SCED can be parameterized by GENCOs' bidding strategies. We apply an active-set method to solve a parametric SCED problem. The idea of implementing the active-set method is based on the fact that the optimal solution of a parametric quadratic programming (QP) problem with linear equality constraints can be determined directly. We also decompose generation scheduling, locational marginal prices, and consequently GENCOs' profits, in terms of GENCOs' bidding strategies. In this situation, the BLP transforms to a single-level inequality-constrained optimization, which maximizes GENCO's profit subject to inactive constraints of the lower level. In addition, GENCOs' strategic interactions in oligopolistic marketplace are investigated by using a game theoretic framework, and optimal bidding strategies are determined then by finding the Nash equilibrium of GENCOs' noncooperative competition. Two case studies are provided to illustrate the proposed method; in addition, a numerical method has been conducted to validate the results. The results clarify the impact of each GENCO's bidding strategy on the power flows of transmission lines.

A novel finite element time domain method for nonlinear Maxwell's equations based on the parametric quadratic programming method

ABSTRACTA finite‐element time‐domain method is developed for Maxwell's equations with nonlinear media. Compared with conventional nonlinear numerical methods, the proposed scheme is not based on iterations, but using the quadratic programming method. The nonlinear constitutive relations are treated as a series of linear complementary problems, thus, this scheme presents very good convergence behavior in computation. Numerical results demonstrate the validity of the proposed method. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:1640–1645, 2015

Missile Guidance Law Based on Robust Model Predictive Control Using Neural-Network Optimization.

In this brief, the utilization of robust model-based predictive control is investigated for the problem of missile interception. Treating the target acceleration as a bounded disturbance, novel guidance law using model predictive control is developed by incorporating missile inside constraints. The combined model predictive approach could be transformed as a constrained quadratic programming (QP) problem, which may be solved using a linear variational inequality-based primal-dual neural network over a finite receding horizon. Online solutions to multiple parametric QP problems are used so that constrained optimal control decisions can be made in real time. Simulation studies are conducted to illustrate the effectiveness and performance of the proposed guidance control law for missile interception.

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.

A Numerical Simulation of Elastoplastic Contact Analysis of Compressor by Overspeeding

This study analyzes the stress distribution of 3D elastoplastic contact problems by using the FE parametric quadratic programming (PQP) method derived from a 3D FE model based on parametric variational principle (PVP). We numerically analyze a 24-blade compressor by combining centrifugal loading with interference-fit one. To accelerate computation, calculation is simplified by structural modeling via multisubstructuring, aiming to deal with FE-simulated computer aided design (CAD) conveniently. We then analyze the relationships between the maximum residual stresses of the compressor posterior to prestressing and overspeed rpms, and we also study the distribution and magnitude of the contact stresses of the compressor in working conditions after overspeed prestressing. Moreover, we thoroughly discuss the distribution and magnitude of the contact stresses of shaft-shaft sleeve-impeller in working conditions. Relative displacement can be prevented and contact stress can be kept uniform due to the nonuniform initial amount of interference in overspeed prestressing. This paper summarizes the FEM simulation results and provides reference data for improving the design and processing of compressor impellers, indicating that overspeed is indispensable in manufacture.

Feasibility and finite convergence analysis for accurate on-line ν-support vector machine.

The ν-support vector machine ( ν-SVM) for classification has the advantage of using a parameter ν on controlling the number of support vectors and margin errors. Recently, an interesting accurate on-line algorithm accurate on-line ν-SVM algorithm (AONSVM) is proposed for training ν-SVM. AONSVM can be viewed as a special case of parametric quadratic programming techniques. It is demonstrated that AONSVM avoids the infeasible updating path as far as possible, and successfully converges to the optimal solution based on experimental analysis. However, because of the differences between AONSVM and classical parametric quadratic programming techniques, there is no theoretical justification for these conclusions. In this paper, we prove the feasibility and finite convergence of AONSVM under two assumptions. The main results of feasibility analysis include: 1) the inverses of the two key matrices in AONSVM always exist; 2) the rules for updating the two key inverse matrices are reliable; 3) the variable ζ can control the adjustment of the sum of all the weights efficiently; and 4) a sample cannot migrate back and forth in successive adjustment steps among the set of margin support vectors, the set of error support vectors, and the set of the remaining vectors. Moreover, the analyses of AONSVM also provide the proofs of the feasibility and finite convergence for accurate on-line C-SVM learning directly.

EXPLICIT MODEL PREDICTIVE CONTROL OF FAST DYNAMIC SYSTEM

Explicit Model Predictive Control approach provides offline computation of the optimization law by Multi Parametric Quadratic Programming. The solution is Piece wise affine in nature. It is explicit representation of the system states and control inputs. Such law then can be solved using binary search tree and can be evaluated for fast dynamic systems. Implementing such controllers can be done on microcontroller or ASIC/FPGA. DC Motor Speed Control - one of the benchmark systems is discussed here in this context. Its PWA law obtained, simulation of closed loop e-MPC is presented and its implementation approach using MPT toolbox and other such toolboxes is shown in brief.

Frictional Contact Problem Study on Compressor by a Parametric Quadratic Programming Method

A locomotive-type turbocharger compressor with 24 blades under combined centrifugal and interference-fit loading was considered in the numerical analysis. The solution of elastoplastic frictional contact problems belongs to the unspecified boundary problems where the interaction between two kinds of nonlinearities should occur. To save the time cost in the numerical computation, multi-substructure technique was adopted in the structural modeling. The effect of fit tolerance, wall thickness of shaft sleeve and rotational speed on the contact stress was discussed in detail in the numerical computation. To decrease the difficulty of the assembling process and make sure the safety of the working state, the amount of interference between the shaft sleeve and shaft by press-fitting should be controlled strictly to avoid the rapid increase of the contact stress. The numerical results show the high accuracy and good convergence of the algorithm presented here. The study play a referenced role in deciding the proper fit tolerance and improving design and manufacturing technology of compressor impellers.

Optimizing fuzzy portfolio selection problems by parametric quadratic programming

This paper develops a robust method to describe fuzzy returns by employing parametric possibility distributions. The parametric possibility distributions are obtained by equivalent value (EV) reduction methods. For common type-2 triangular and trapezoidal fuzzy variables, their reduced fuzzy variables are studied in the current development. The parametric possibility distributions of reduced fuzzy variables are first derived, then the second moment formulas for the reduced fuzzy variables are established. Taking the second moment as a new risk measure, the reward-risk and risk-reward models are developed to optimize fuzzy portfolio selection problems. The mathematical properties of the proposed optimization models are analyzed, including the analytical representations for the second moments of linear combinations of reduced fuzzy variables as well as the convexity of second moments with respect to decision vectors. On the basis of the analytical representations for the second moments, the reward-risk and risk-reward models can be turned into their equivalent parametric quadratic convex programming problems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling ideas and the efficiency of solution method.

Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.

Inverse Parametric Quadratic Programming and an Application to Hybrid Control

We present the complete solution to the inverse parametric quadratic programming problem: from a given continuous piecewise affine function we construct both the constraints and objective function of a parametric quadratic program, such that the supplied function is the unique parametric minimizer for the constructed problem data. In contrast to past approaches to this problem, our method does not rely on prior knowledge of the constraint set or sufficient sampling of the optimizer function, and is guaranteed to solve the inverse optimization problem exactly if a solution exists. We then apply this inverse optimization technique to the control of piecewise affine systems. By recasting the hybrid system dynamics as the parametric solution to a quadratic program obtained from our inverse optimization technique, we derive an equivalent linear complementarity model via the Karush-Kuhn-Tucker conditions of the identified optimization problem. This approach allows one to solve an optimal control problem for a piecewise affine system by solving a mathematical program with equilibrium constraints. Simulation results suggest that the computational effort required to solve such problems can be significantly smaller than that required for conventional mixed-integer quadratic programming approaches for systems with piecewise affine dynamics. We demonstrate via two numerical examples that globally optimal points can be identified using this approach.