Rank-One Scaled ℋ∞ Optimization

1997 ◽  
Vol 119 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Tetsuya Iwasaki ◽  
Mario A. Rotea
Keyword(s):  
Flight Control ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Design Tool ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Convex Optimization Problems ◽  
Rank One ◽  
Plant Transfer ◽  
Primary Contribution

This paper gives a solution to the scaled ℋ∞ optimization problem with constant scalings and output feedback. It is shown that, when the nominal plant transfer matrix has a special rank-one property, this optimization problem is equivalent to a sequence of convex optimization problems involving linear matrix inequalities. These results are demonstrated with a flight control example. The primary contribution of the example is a method for weight selection that is applicable to problems in which ℋ∞ optimization is used as the design tool.

scholarly journals Multivariate Spectral Gradient Algorithm for Nonsmooth Convex Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Yaping Hu
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Nonsmooth Convex Optimization ◽  
Convex Optimization Problem ◽  
Convex Optimization Problems ◽  
Yosida Regularization ◽  
Approximate Function ◽  
Original Objective

We propose an extended multivariate spectral gradient algorithm to solve the nonsmooth convex optimization problem. First, by using Moreau-Yosida regularization, we convert the original objective function to a continuously differentiable function; then we use approximate function and gradient values of the Moreau-Yosida regularization to substitute the corresponding exact values in the algorithm. The global convergence is proved under suitable assumptions. Numerical experiments are presented to show the effectiveness of this algorithm.

scholarly journals Sum-of-squares chordal decomposition of polynomial matrix inequalities

2021 ◽  
Author(s):  
Yang Zheng ◽  
Giovanni Fantuzzi
Keyword(s):  
Optimization Problems ◽  
Polynomial Matrix ◽  
Positive Semidefinite ◽  
Positive Definite ◽  
Sum Of Squares ◽  
Matrix Inequalities ◽  
Representation Theorems ◽  
Convex Optimization Problems ◽  
Sparse Polynomial ◽  
Decomposition Theorems

AbstractWe prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all $$x\in \mathbb {R}^n$$ x ∈ R n if and only if there exists a sum-of-squares (SOS) polynomial $$\sigma (x)$$ σ ( x ) such that $$\sigma P$$ σ P is a sum of sparse SOS matrices. Second, we show that setting $$\sigma (x)=(x_1^2 + \cdots + x_n^2)^\nu $$ σ ( x ) = ( x 1 2 + ⋯ + x n 2 ) ν for some integer $$\nu $$ ν suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set $$\mathcal {K}=\{x:g_1(x)\ge 0,\ldots ,g_m(x)\ge 0\}$$ K = { x : g 1 ( x ) ≥ 0 , … , g m ( x ) ≥ 0 } satisfying the Archimedean condition, then $$P(x) = S_0(x) + g_1(x)S_1(x) + \cdots + g_m(x)S_m(x)$$ P ( x ) = S 0 ( x ) + g 1 ( x ) S 1 ( x ) + ⋯ + g m ( x ) S m ( x ) for matrices $$S_i(x)$$ S i ( x ) that are sums of sparse SOS matrices. Finally, if $$\mathcal {K}$$ K is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for $$(x_1^2 + \cdots + x_n^2)^\nu P(x)$$ ( x 1 2 + ⋯ + x n 2 ) ν P ( x ) with some integer $$\nu \ge 0$$ ν ≥ 0 when P and $$g_1,\ldots ,g_m$$ g 1 , … , g m are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.

scholarly journals A Fixed-Point Subgradient Splitting Method for Solving Constrained Convex Optimization Problems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 377
Author(s):  
Nimit Nimana
Keyword(s):  
Fixed Point ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Splitting Method ◽  
Optimal Solution ◽  
Bilevel Optimization ◽  
Convergence Properties ◽  
Nonlinear Operators ◽  
Convex Optimization Problems ◽  
Feasible Points

In this work, we consider a bilevel optimization problem consisting of the minimizing sum of two convex functions in which one of them is a composition of a convex function and a nonzero linear transformation subject to the set of all feasible points represented in the form of common fixed-point sets of nonlinear operators. To find an optimal solution to the problem, we present a fixed-point subgradient splitting method and analyze convergence properties of the proposed method provided that some additional assumptions are imposed. We investigate the solving of some well known problems by using the proposed method. Finally, we present some numerical experiments for showing the effectiveness of the obtained theoretical result.

Large Scale Structural Optimization With Linear Goal Programming

1990 ◽  
Author(s):  
Mohamed E. M. El-Sayed ◽  
T. S. Jang
Keyword(s):  
Structural Optimization ◽  
Goal Programming ◽  
Large Scale ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Minimum Weight ◽  
Design Tool ◽  
Linear Goal Programming ◽  
Programming Techniques ◽  
Structural Optimization Problem

Abstract This paper presents a method for solving large scale structural optimization problems using linear goal programming techniques. The method can be used as a multicriteria optimization tool since goal programming removes the difficulty of having to define an objective function and constraints. It also has the capacity of handling rank ordered design objectives or goals. The method uses finite element analysis, linear goal programming techniques and successive linearization to obtain the solution for the nonlinear goal optimization problems. The general formulation of the structural optimization problem into a nonlinear goal programming form is presented. The successive linearization method for the nonlinear goal optimization problem is discussed. To demonstrate the validity of the method, as a design tool, the solution of the minimum weight structural optimization problem with stress constraints for 10, 25 and 200 truss problems are included.

Structural Optimization With Nonlinear Goal Programming Using Powell’s Method

1991 ◽  
Author(s):  
Mohamed E. M. El-Sayed ◽  
T. S. Jang
Keyword(s):  
Structural Optimization ◽  
Goal Programming ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Numerical Test ◽  
Design Tool ◽  
Test Cases ◽  
Programming Techniques ◽  
Ordered Design

Abstract This paper presents a method for solving structural optimization problems using nonlinear goal programming techniques. The developed method removes the difficulty of having to define an objective function and constraints. It also has the capacity of handling rank ordered design objectives or goals. The formulation of the structural optimization problem into a goal programming form is discussed. The resulting optimization problem is solved using Powell’s conjugate direction search algorithm. To demonstrate the effectiveness of the method, as a design tool, the solutions of some numerical test cases are included.

Counterweight Balancing for Reducing Machine Frame Vibration: A Convex Optimization Framework

2006 ◽  
Author(s):  
Myriam Verschuure ◽  
Bram Demeulenaere ◽  
Jan Swevers ◽  
Joris De Schutter
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Global Optimum ◽  
Convex Optimization Problem ◽  
Optimization Framework ◽  
Convex Optimization Problems ◽  
Drive Speed ◽  
Maximal Reduction

This paper focusses on reducing, through counterweight addition, the vibration of an elastically mounted, rigid machine frame that supports a linkage. In order to determine the counterweights that yield a maximal reduction in frame vibration, a non-linear optimization problem is formulated with the frame kinetic energy as objective function and such that a convex optimization problem is obtained. Convex optimization problems are nonlinear optimization problems that have a unique (global) optimum, which can be found with great efficiency. The proposed methodology is successfully applied to improve the results of the benchmark four-bar problem, first considered by Kochev and Gurdev. For this example, the balancing is shown to be very robust for drive speed variations and to benefit only marginally from using a coupler counterweight.

scholarly journals Pole Assignment for Active Vibration Control of Linear Vibrating Systems through Linear Matrix Inequalities

2020 ◽  
Vol 10 (16) ◽  
pp. 5494 ◽  
Author(s):  
Roberto Belotti ◽  
Dario Richiedei ◽  
Iacopo Tamellin ◽  
Alberto Trevisani
Keyword(s):  
Linear Matrix Inequalities ◽  
Complex Plane ◽  
Degrees Of Freedom ◽  
Active Vibration Control ◽  
Pole Placement ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Rank One ◽  
Uniqueness Of The Solution ◽  
Vibrating Systems

This paper proposes a novel method for pole placement in linear vibrating systems through state feedback and rank-one control. Rather than assigning all the poles to the desired locations of the complex plane, the proposed method exactly assigns just the dominant poles, while the remaining ones are free to assume arbitrary positions within a pre-specified region in the complex plane. Therefore, the method can be referred to as “regional pole placement”. A two-stage approach is proposed to accomplish both the tasks. In the first stage, the subset of dominant poles is assigned to exact locations by exploiting the receptance method, formulated for either symmetric or asymmetric systems. Then, in the second stage, a first-order model formulated with a reduced state, together with the theory of Linear Matrix Inequalities, are exploited to cluster the subset of the unassigned poles into some stable regions of the complex plane while keeping unchanged the poles assigned in the first stage. The additional degrees of freedom in the choice of the gains, i.e., the non-uniqueness of the solution, is exploited through a semidefinite programming problem to reduce the control gains. The method is validated by means of four meaningful and challenging test-cases, also borrowed from the literature. The results are also compared with those of classic partial pole placement, to show the benefits and the effectiveness of the proposed approach.

scholarly journals Joint Design of Colocated MIMO Radar Constant Envelope Waveform and Receive Filter to Reduce SINR Loss

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3887
Author(s):  
Liang Huang ◽  
Xiaofang Deng ◽  
Lin Zheng ◽  
Huiping Qin ◽  
Hongbing Qiu
Keyword(s):  
Fractional Programming ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Design Method ◽  
Solution Process ◽  
Mimo Radar ◽  
Joint Design ◽  
Convex Optimization Problems ◽  
Constant Envelope ◽  
Output Sinr

In this paper, we aim at the problem that MIMO radar’s target detection performance is greatly reduced in the complex multi-signal-dependent interferences environment. We propose a joint design method based on semidefinite relaxation (SDR), fractional programming and randomization technique (JD-SFR) and a joint design method based on coordinate descent (JD-CD) to solve the actual transmit waveform and receive filter bank directly to reduce the loss of strong interference to the output signal-to-interference-plus-noise ratio (SINR) of the radar system. Therefore, the maximization of output SINR is taken as the criterion of the optimization problem. The designed waveforms take into account the radar transmitter’s hardware requirements for constant envelope waveforms and impose similarity constraints on the waveforms. JD-SFR uses SDR, fractional programming and randomization technique to deal with the non-convex optimization problems encountered in the solution process. JD-CD transforms the optimization problem into a function of the phase of the waveform and then solves the transmit waveform based on CD. Compared with other methods, the proposed method has lower output SINR loss under strong power interference and forms deep nulls on the direction beampattern of multiple interference sources, which indicates that it has better anti-interference performance.

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