Chen-Harker-Kanzow-Smale Research Articles

A note on the dislocation hyperbolic transformation function

In work of Roman A. Polyak [R.A. Polyak, Math. Program. 92 (2002) 197–235.], the Modified Chen-Harker-Kanzow-Smale (CHKS) function was studied to relate a multiplier method and an Interior Prox method with the second order distance function. Independently, the dislocated hyperbolic penalty function (DHPF) was proposed by A.E. Xavier (1992). DHPF was rewritten and studied in [L. Mallma Ramirez, N. Maculan, A.E. Xavier and V.L. Xavier, RAIRO:RO 57 (2023) 2941–2950.] and [L. Mallma Ramirez, N. Maculan, A.E. Xavier and V.L. Xavier, J. Convex Anal. To appear (2024).]. Thus, this function was called the dislocated hyperbolic function (DHF). In this work, we note that DHF is a particular case of CHKS function. Then we will call the DHF function as the dislocation hyperbolic transformation function.

A nonmonton active interior point trust region algorithm based on CHKS smoothing function for solving nonlinear bilevel programming problems

<abstract><p>In this paper, an approach is suggested to solve nonlinear bilevel programming (NBLP) problems. In the suggested method, we convert the NBLP problem into a standard nonlinear programming problem with complementary constraints by applying the Karush-Kuhn-Tucker condition to the lower-level problem. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the nonlinear programming problem is successively smoothed. A nonmonton active interior-point trust-region algorithm is introduced to solve the smoothed nonlinear programming problem to obtain an approximately optimal solution to the NBLP problem. Results from simulations on several benchmark problems and a real-world case about a watershed trading decision-making problem show how the effectiveness of the suggested approach in NBLP solution development.</p></abstract>

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.

A Binary PSO-Based Model Selection for Novel Smooth Twin Support Vector Regression

The recently proposed smooth twin support vector regression, denoted by STSVR, gains better training speed compared with twin support vector regression (TSVR). In the STSVR, sigmoid function is used for the smooth function, however, its approximation precision is relatively low, leading to the generalization performance of STSVR is not good enough. Moreover, STSVR has at least three parameters that need regulating, which affects its practical applications. In this paper, we increase the regression performance of STSVR from two aspects. First, by introducing Chen-Harker-Kanzow-Smale (CHKS) function, a new smooth version for TSVR, termed as smooth CHKS twin support vector regression (SCTSVR) is proposed. Second, a binary particle swarm optimization (PSO)-based model selection for SCTSVR is suggested. Computational results on one synthetic as well as several benchmark datasets confirm the great improvements on the training process of proposed algorithm.

New convergence analysis of a class of smoothing Newton-type methods for second-order cone complementarity problem

<abstract><p>In this paper we propose a class of smoothing Newton-type methods for solving the second-order cone complementarity problem (SOCCP). The proposed method design is based on a special regularized Chen-Harker-Kanzow-Smale (CHKS) smoothing function. When the solution set of the SOCCP is nonempty, our method has the following convergence properties: (ⅰ) it generates a bounded iteration sequence; (ⅱ) the value of the merit function converges to zero; (ⅲ) any accumulation point of the generated iteration sequence is a solution of the SOCCP; (ⅳ) it has the local quadratic convergence rate under suitable assumptions. Some numerical results are reported.</p></abstract>

Nonlinear Model Predictive Tracking Control With C/GMRES Method for Heavy-Duty AGVs

To ensure trajectory tracking accuracy and driving safety, this paper presents a real time nonlinear model predictive control (NMPC) method for autonomous guided vehicles (AGVs) to avoid dangerous load transfer effect in carrying heavy equipment. The AGV with a four-wheel-steering four-wheel-driving (4WS4WD) structure is controlled with a steer and velocity input for each wheel. The tracking problem is implemented by the standard NMPC framework and the referenced trajectory is planned by the Reeds-Shepp algorithm. A Large load creates a considerable load transfer effect which is dangerous for the wheels; hence, the controller limits the lateral acceleration to reduce load transfer. To realize online numerical computation for NMPC, the continuation/generalized minimal residual (C/GMRES) method is applied. A semismooth transformation based on the Chen-Harker-Kanzow-Smale (CHKS) function is introduced to handle inequality constraints since C/GMRES only solves standard equality forms. Trajectory tracking simulations are carried out in contrast to the quadratic programming method without lateral load transfer optimization, and practical experiments on an AGV carrying 13 tons of steel are also performed. The results indicate the accuracy, efficiency, safety and applicability of the proposed method on heavy-duty AGVs.

Optimal Pressure Management for Large-Scale Water Distribution Systems Using Smoothing Model

Optimal pressure management in water distribution systems (WDSs) is one of the most efficient approaches to control water leakage for water utilities worldwide. The optimal pressure management can be accomplished through regulating operations of pressure reducing valves (PRVs) to ensure that the excessive pressure in the WDS is minimized. This engineering task can be casted into a nonlinear program problem (NLP) with non-smooth constraints. Until now, the non-smooth constraints have been approximated by the smoothing function of Chen Harker-Kanzow-Smale (CHKS). In this paper, instead of using the CHKS function, we propose to apply the uniform smoothing function for formulation of the NLP. Numerical simulations using two smoothing functions will be carried out for optimal pressure managements of a benchmark WDS and a real-world WDS in Thainguyen City, in Vietnam. The comparison results reveal that the NLP formulated with the uniform smoothing function outperforms the existing NLP formulated with the CHKS in terms of optimal solution accuracy.

Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints

A stochastic mathematical program model with second-order cone complementarity constraints (SSOCMPCC) is introduced in this paper. It can be considered as a non-trivial extension of stochastic mathematical program with complementarity constraints, and could arise from a hard-to-handle class of bilivel second-order cone programming and inverse stochastic second-order cone programming. By introducing the Chen-Harker-Kanzow-Smale (CHKS) type function to replace the projection operator onto the second-order cone, a smoothing sample average approximation (SAA) method is proposed for solving the SSOCMPCC problem. It can be shown that with proper assumptions, as the sample size goes to infinity, any cluster point of global solutions of the smoothing SAA problem is a global solution of SSOCMPCC almost surely, and any cluster point of stationary points of the former problem is a C-stationary point of the latter problem almost surely. C-stationarity can be strengthened to M-stationarity with additional assumptions. Finally, we report a simple illustrative numerical test to demonstrate our theoretical results.

The locally Chen–Harker–Kanzow–Smale smoothing functions for mixed complementarity problems

According to the structure of the projection function onto the box set $$\varPi _X$$ and the Chen–Harker–Kanzow–Smale (CHKS) smoothing function, a new class of smoothing projection functions onto the box set are proposed in this paper. The new smoothing projection functions only smooth $$\varPi _X$$ in neighborhoods of nonsmooth points of $$\varPi _X$$ , and keep unchanged with $$\varPi _X$$ at other points, hence they are referred as the locally Chen–Harker–Kanzow–Smale (LCHKS) smoothing functions. Based on the Robinson’s normal equation and the LCHKS smoothing functions, a smoothing Newton method with its convergence results is proposed for solving mixed complementarity problems. Compared with smoothing Newton methods based on various smoothing projection functions, the computations of the LCHKS smoothing function, the function value and its Jacobian matrix of the Newton equation become cheaper, and the Newton direction can be found by solving a low dimensional linear equation, hence the smoothing Newton method based on the LCHKS smoothing functions shows more efficient for large-scale mixed complementarity problems. The LCHKS smoothing functions are proved to be feasible, continuously differentiable, uniform approximations of $$\varPi _X$$ , globally Lipschitz continuous and strongly semismooth, which are important to establish the superlinear and quadratic convergence of the smoothing Newton method. The proposed smoothing Newton method is implemented in MATLAB and numerical tests are done on the MCPLIB test collection. Numerical results show that the smoothing Newton method based on the LCHKS smoothing functions is promising for mixed complementarity problems.

A smoothing Levenberg-Marquardt method for nonlinear complementarity problems

As is well-known, Jacobian smoothing method is a popular one to solve nonlinear complementarity problems, in which the Jacobian consistency is stressed. By investigating an element of related functions’ B-differential, a smoothing Levenberg-Marquardt(LM) method is proposed based on a Chen-Harker-Kanzow-Smale(CHKS) smoothing function, which satisfies a property called strongly Jacobian consistency. Finally, the numerical experiments illustrate the effectiveness of the given method.

Modulation recognition of communication signals based on SCHKS-SSVM

A novel modulation recognition algorithm is proposed by introducing a Chen-Harker-Kanzow-Smale (CHKS) smooth function into the C-support vector machine deformation algorithm. A set of seven characteristic parameters is selected from a range of parameters of communication signals including instantaneous amplitude, phase, and frequency. And the Newton-Armijo algorithm is utilized to train the proposed algorithm, namely, smooth CHKS smooth support vector machine (SCHKS-SSVM). Compared with the existing algorithms, the proposed algorithm not only solves the non-differentiable problem of the second order objective function, but also reduces the recognition error. It significantly improves the training speed and also saves a large amount of storage space through large-scale sorting problems. The simulation results show that the recognition rate of the algorithm can batch training. Therefore, the proposed algorithm is suitable for solving the problem of high dimension and its recognition can exceed 95% when the signal-to-noise ratio is no less than 10 dB.

A new smoothing and regularization Newton method for the symmetric cone complementarity problem

Based on a regularized Chen–Harker–Kanzow–Smale (CHKS) smoothing function, we propose a new smoothing and regularization Newton method for solving the symmetric cone complementarity problem. By using the theory of Euclidean Jordan algebras, we establish the global and local quadratic convergence of the method on certain assumptions. The proposed method uses a nonmonotone line search technique which includes the usual monotone line search as a special case. In addition, our method treats both the smoothing parameter $$\mu $$ and the regularization parameter $$\varepsilon $$ as independent variables. Preliminary numerical results are reported which indicate that the proposed method is effective.

An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bilevel programming problems

Bilevel programming techniques deal with decision processes involving two decision makers with a hierarchical structure. In this paper, an augmented Lagrangian multiplier method is proposed to solve nonlinear bilevel programming (NBLP) problems. An NBLP problem is first transformed into a single level problem with complementary constraints by replacing the lower level problem with its Karush–Kuhn–Tucker optimality condition, which is sequentially smoothed by a Chen–Harker–Kanzow–Smale (CHKS) smoothing function. An augmented Lagrangian multiplier method is then applied to solve the smoothed nonlinear program to obtain an approximate optimal solution of the NBLP problem. The asymptotic properties of the augmented Lagrangian multiplier method are analyzed and the condition for solution optimality is derived. Numerical results showing viability of the approach are reported.

Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bilevel programming problem

Particle Swarm Optimization (PSO) is a new optimization technique originating from artificial life and evolutionary computation. It completes optimization through following the personal best solution of each particle and the global best value of the whole swarm. PSO can be used to solve nonlinear programming problems for global optimal solutions efficiently, so a novel approach based on particle swarm optimization is proposed to solve nonlinear bilevel programming problem (NBLP). In the proposed approach, applying Karush–Kuhn–Tucker (KKT) condition to the lower level problem, we transform the NBLP into a regular nonlinear programming with complementary constraints, which is sequentially smoothed by Chen-Harker-Kanzow-Smale (CHKS) smoothing function. The PSO approach is then applied to solve the smoothed nonlinear programming for getting the approximate optimal solution of the NBLP problem. Simulations on 5 benchmark problems and practical example about watershed water trading decision-making problem are made and the results demonstrate the effectiveness of the proposed method for solving NBLP.

An % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % oe-taaa!44C5! $$ \mathcal{O} $$ (rL) infeasible interior-point algorithm for symmetric cone LCP via CHKS function

In this paper, we propose a theoretical framework of an infeasible interior-point algorithm for solving monotone linear complementarity problems over symmetric cones (SCLCP). The new algorithm gets Newton-like directions from the Chen-Harker-Kanzow-Smale (CHKS) smoothing equation of the SCLCP. It possesses the following features: The starting point is easily chosen; one approximate Newton step is computed and accepted at each iteration; the iterative point with unit stepsize automatically remains in the neighborhood of central path; the iterative sequence is bounded and possesses \( \mathcal{O} \)(rL) polynomial-time complexity under the monotonicity and solvability of the SCLCP.

Analysis of a non-interior continuation method for second-order cone programming

Based on the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, a non-interior continuation method is presented for solving the second-order cone programming (SOCP). Our algorithm reformulates the SOCP as a nonlinear system of equations and then applies Newton’s method to the perturbation of this system. The proposed algorithm does not have restrictions regarding its starting point and solves at most one linear system of equations at each iteration. Under suitable assumptions, the algorithm is shown to be globally and locally quadratically convergent. Some numerical results are also included which indicate that our algorithm is promising and comparable to interior-point methods.

Sub-quadratic convergence of a smoothing Newton method for second-order cone programming

In this paper, we present a smoothing Newton method for solving the second-order cone programming (SOCP) based on the Chen–Harker–Kanzow–Smale (CHKS) smoothing function. Our smoothing method reformulates SOCP as a nonlinear system of equations and then applies Newton’s method to the system. The proposed method solves only one linear system of equations and performs only one line search at each iteration. It is shown that the method is globally and locally sub-quadratically convergent under a nonsingularity assumption. Numerical results suggest that the method is promising.

Sub-quadratic convergence of a smoothing Newton algorithm for the P 0 ? and monotone LCP

Given *** equation here ***, the linear complementarity problem (LCP) is to find *** equation here *** such that (x, s)≥ 0,s=Mx+q,xTs=0. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the LCP is reformulated as a system of parameterized smooth-nonsmooth equations. As a result, a smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm [Mathematical Programming, Vol. 87, 2000, pp. 1–35], is proposed to solve the LCP with M being assumed to be a P0-matrix (P0–LCP). The proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration. It is proved in this paper that the proposed algorithm has the following convergence properties: (i) it is well-defined and any accumulation point of the iteration sequence is a solution of the P0–LCP; (ii) it generates a bounded sequence if the P0–LCP has a nonempty and bounded solution set; (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P0–LCP has a unique solution, then the whole iteration sequence converges to this accumulation point sub-quadratically with a Q-rate 2–t, where t∈(0,1) is a parameter; and (iv) if M is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition, then the whole sequence converges to the accumulation point quadratically.