scholarly journals AUV-Method for a Class of Constrained Minimized Problems of Maximum Eigenvalue Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Wei Wang ◽  
Ming Jin ◽  
Shanghua Li ◽  
Xinyu Cao
Keyword(s):  
Exact Penalty ◽  
Maximum Eigenvalue ◽  
Convex Approximation ◽  
Equivalent Problem ◽  
Composite Function ◽  
Primal Problem ◽  
Approximate Problem ◽  
Constrained Problem ◽  
Constrained Minimization Problem ◽  
Unconstrained Problem

In this paper, we apply theUV-algorithm to solve the constrained minimization problem of a maximum eigenvalue function which is the composite function of an affine matrix-valued mapping and its maximum eigenvalue. Here, we convert the constrained problem into its equivalent unconstrained problem by the exact penalty function. However, the equivalent problem involves the sum of two nonsmooth functions, which makes it difficult to applyUV-algorithm to get the solution of the problem. Hence, our strategy first applies the smooth convex approximation of maximum eigenvalue function to get the approximate problem of the equivalent problem. Then the approximate problem, the space decomposition, and theU-Lagrangian of the object function at a given point will be addressed particularly. Finally, theUV-algorithm will be presented to get the approximate solution of the primal problem by solving the approximate problem.

A Proximal Bundle Method with Exact Penalty Technique and Bundle Modification Strategy for Nonconvex Nonsmooth Constrained Optimization

2021 ◽  
pp. 2150015
Author(s):  
Xiaoliang Wang ◽  
Liping Pang ◽  
Qi Wu
Keyword(s):  
Operation Research ◽  
Bundle Method ◽  
Exact Penalty ◽  
Constraint Function ◽  
Modified Model ◽  
Proximal Bundle Method ◽  
Unconstrained Problem ◽  
Unconstrained Problems ◽  
Penalty Strategy ◽  
Penalty Technique

The bundle modification strategy for the convex unconstrained problems was proposed by Alexey et al. [[2007] European Journal of Operation Research, 180(1), 38–47.] whose most interesting feature was the reduction of the calls for the quadratic programming solver. In this paper, we extend the bundle modification strategy to a class of nonconvex nonsmooth constraint problems. Concretely, we adopt the convexification technique to the objective function and constraint function, take the penalty strategy to transfer the modified model into an unconstrained optimization and focus on the unconstrained problem with proximal bundle method and the bundle modification strategies. The global convergence of the corresponding algorithm is proved. The primal numerical results show that the proposed algorithms are promising and effective.

scholarly journals A Relax Inexact Accelerated Proximal Gradient Method for the Constrained Minimization Problem of Maximum Eigenvalue Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Wei Wang ◽  
Shanghua Li ◽  
Jingjing Gao
Keyword(s):  
Minimization Problem ◽  
Feasible Solution ◽  
Lower Semicontinuous ◽  
Constrained Minimization ◽  
Maximum Eigenvalue ◽  
Proximal Gradient Method ◽  
Smooth Approximation ◽  
Computational Advantage ◽  
Global Iteration ◽  
Constrained Minimization Problem

For constrained minimization problem of maximum eigenvalue functions, since the objective function is nonsmooth, we can use the approximate inexact accelerated proximal gradient (AIAPG) method (Wang et al., 2013) to solve its smooth approximation minimization problem. When we take the functiong(X)=δΩ(X)  (Ω∶={X∈Sn:F(X)=b,X⪰0})in the problemmin{λmax(X)+g(X):X∈Sn}, whereλmax(X)is the maximum eigenvalue function,g(X)is a proper lower semicontinuous convex function (possibly nonsmooth) andδΩ(X)denotes the indicator function. But the approximate minimizer generated by AIAPG method must be contained inΩotherwise the method will be invalid. In this paper, we will consider the case where the approximate minimizer cannot be guaranteed inΩ. Thus we will propose two different strategies, respectively, constructing the feasible solution and designing a new method named relax inexact accelerated proximal gradient (RIAPG) method. It is worth mentioning that one advantage when compared to the former is that the latter strategy can overcome the drawback. The drawback is that the required conditions are too strict. Furthermore, the RIAPG method inherits the global iteration complexity and attractive computational advantage of AIAPG method.

scholarly journals Fuel-Optimal Ascent Trajectory Problem for Launch Vehicle via Theory of Functional Connections

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Shiyao Li ◽  
Yushen Yan ◽  
Kun Zhang ◽  
Xinguo Li
Keyword(s):  
Optimization Problem ◽  
Search Space ◽  
Launch Vehicle ◽  
Functional Connections ◽  
Constrained Problem ◽  
Unconstrained Optimization Problem ◽  
Interpolation Technique ◽  
Matlab Implementation ◽  
Unconstrained Problem ◽  
Trajectory Problem

In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in the ascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem (TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’s constraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimization problem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space of the solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknown coefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm is validated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numerical method. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved in less than 300 ms within the MATLAB implementation. Consequently, the proposed method has the potential to generate optimal trajectories on-board in real time.

scholarly journals Smooth Convex Approximation to the Maximum Eigenvalue Function

2004 ◽  
Vol 30 (2-3) ◽  
pp. 253-270 ◽  
Author(s):  
Xin Chen ◽  
Houduo Qi ◽  
Liqun Qi ◽  
Kok-Lay Teo
Keyword(s):  
Smooth Convex ◽  
Maximum Eigenvalue ◽  
Convex Approximation

scholarly journals Distributionally Robust Joint Chance Constrained Problem under Moment Uncertainty

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ke-wei Ding
Keyword(s):  
Semidefinite Programming ◽  
Chance Constraints ◽  
Second Order ◽  
Quadratic Approximation ◽  
Convex Approximation ◽  
Worst Case ◽  
Constrained Problem ◽  
Distributionally Robust

We discuss and develop the convex approximation for robust joint chance constraints under uncertainty of first- and second-order moments. Robust chance constraints are approximated by Worst-Case CVaR constraints which can be reformulated by a semidefinite programming. Then the chance constrained problem can be presented as semidefinite programming. We also find that the approximation for robust joint chance constraints has an equivalent individual quadratic approximation form.

scholarly journals Left and Right Inverse Eigenpairs Problem forκ-Hermitian Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Fan-Liang Li ◽  
Xi-Yan Hu ◽  
Lei Zhang
Keyword(s):  
Approximate Solution ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Optimal Approximate Solution ◽  
Hermitian Matrices ◽  
Equivalent Problem ◽  
Approximate Problem ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Product Of Matrices

Left and right inverse eigenpairs problem forκ-hermitian matrices and its optimal approximate problem are considered. Based on the special properties ofκ-hermitian matrices, the equivalent problem is obtained. Combining a new inner product of matrices, the necessary and sufficient conditions for the solvability of the problem and its general solutions are derived. Furthermore, the optimal approximate solution and a calculation procedure to obtain the optimal approximate solution are provided.

scholarly journals New Exact Penalty Functions for Nonlinear Constrained Optimization Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Bingzhuang Liu ◽  
Wenling Zhao
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Penalty Functions ◽  
Exact Penalty Functions ◽  
Local Minimizers ◽  
Constrained Optimization Problems ◽  
Nonlinear Constrained Optimization ◽  
Primal Problem ◽  
Mild Conditions ◽  
Constrained Problem

For two kinds of nonlinear constrained optimization problems, we propose two simple penalty functions, respectively, by augmenting the dimension of the primal problem with a variable that controls the weight of the penalty terms. Both of the penalty functions enjoy improved smoothness. Under mild conditions, it can be proved that our penalty functions are both exact in the sense that local minimizers of the associated penalty problem are precisely the local minimizers of the original constrained problem.

scholarly journals A parameter-free unconstrained reformulation for nonsmooth problems with convex constraints

2021 ◽  
Author(s):  
Giulio Galvan ◽  
Marco Sciandrone ◽  
Stefano Lucidi
Keyword(s):  
Optimization Methods ◽  
Local Minima ◽  
Convex Constraints ◽  
Nonsmooth Problems ◽  
Constrained Problem ◽  
Nonsmooth Problem ◽  
Feasible Sets ◽  
Unconstrained Problem ◽  
Global And Local ◽  
State Of Art

AbstractIn the present paper we propose to rewrite a nonsmooth problem subjected to convex constraints as an unconstrained problem. We show that this novel formulation shares the same global and local minima with the original constrained problem. Moreover, the reformulation can be solved with standard nonsmooth optimization methods if we are able to make projections onto the feasible sets. Numerical evidence shows that the proposed formulation compares favorably against state-of-art approaches. Code can be found at https://github.com/jth3galv/dfppm.

scholarly journals Power Control for SISO Interference Channel Networks Based on Successive Convex Approximation

2020 ◽  
Author(s):  
Ruina Mao ◽  
Jiguo Yu ◽  
Anming Dong ◽  
Yue Wang ◽  
Baogui Huang
Keyword(s):  
Power Control ◽  
Control Problem ◽  
Taylor Expansion ◽  
Maximization Problem ◽  
Feasible Region ◽  
Interference Channel ◽  
Convex Approximation ◽  
Successive Convex Approximation ◽  
Unconstrained Problem ◽  
The One

Abstract Interference channel (IC) is a classical model used to characterize the effect of interference for many real-life communication systems. In this paper, we consider a sum rate maximization problem subject to maximum power restriction at each transmitter for a single-input single-output (SISO) IC network. The considered power control problem is typically a nonconvex problem which is hard to solve directly. We propose a solving algorithm for this power control problem based on successive convex approximation (SCA). Specifically, we first reformulate the original nonconvex objective function as the difference of two concave (D.C.) functions near a given point in the feasible region of the solution space. After that, we construct a convex substitute function for the D.C. objective by approximating its second concave function with the one-order Taylor expansion. The problem is further reformulated as an unconstrained problem by transforming the constraints as a barrier function. This unconstrained problem is then solved iteratively using Newton's method. We update the substitute function given the newly obtained solution, near which the one-order Taylor expansion is performed again. This process is then repeated until a smooth point is reached. Simulation results show the effectiveness of the proposed SCA-based power control algorithm.

Sign in / Sign up

Export Citation Format

Share Document