Predictor-corrector interior-point algorithm for linearly constrained convex programming

Active set method and gradient projection method are curre nt ly the main approaches for linearly constrained convex programming. Interior-po int method is one of the most effective choices for linear programming. In the p aper a predictor-corrector interior-point algorithm for linearly constrained c onvex programming under the predictor-corrector motivation was proposed. In eac h iteration, the algorithm first performs a predictor-step to reduce the dualit y gap and then a corrector-step to keep the points close to the central traject ory. Computations in the algorithm only require that the initial iterate be nonn egative while feasibility or strict feasibility is not required. It is proved th at the algorithm is equivalent to a level-1 perturbed composite Newton method. Numerical experiments on twenty-six standard test problems are made. The result s show that the proposed algorithm is stable and robust.

Stochastic level-value approximation for quadratic integer convex programming

We propose a stochastic level value approximation method for a quadratic integer convex minimizing problem in this paper. This method applies an importance sampling technique, and make use of the cross-entropy method to update the sample density functions. We also prove the asymptotic convergence of this algorithm, and report some numerical results to illuminate its effectiveness.

A POTENTIAL REDUCTION ALGORITHM FOR LINEARLY CONSTRAINED CONVEX PROGRAMMING

A potential reduction algorithm is proposed for optimization of a convex function subject to linear constraints.At each step of the algorithm,a system of linear equations is solved to get a search direction and the Armijo's rule is used to determine a stepsize.It is proved that the algorithm is globally convergent.Computational results are reported.

NEWTON METHOD FOR SOLVING A CLASS OF SMOOTH CONVEX PROGRAMMING

An algorithm for solving a class of smooth convex programming is given. Using smooth exact multiplier penalty function, a smooth convex programming is minimized to a minimizing strongly convex function on the compact set was reduced. Then the strongly convex function with a Newton method on the given compact set was minimized.

PARALLEL MULTIPLICATIVE ITERATIVE METHODS FOR CONVEX PROGRAMMING

In this paper, we present two parallel multiplicative algorithms for convex programming. If the objective function has compact level sets and has a locally Lipschitz continuous gradient, we discuss convergence of the algorithms. The proofs are essentially based on the results of sequential methods shown by Eggermontt[1].

The Second-Order Differential Equation System with the Feedback Controls for Solving Convex Programming

In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent operator equations for the convex programming problems under the certain conditions are obtained. Then a second-order differential equation system with the feedback controls is constructed on the basis of operator equation. We prove that any accumulation point of the trajectory of the second-order differential equation system with the feedback controls is a solution to the convex programming problem. In the end, two examples using this differential equation system are solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the feedback controls for solving the convex programming problem.

Incorrect Results for E-Convex Functions and E-Convex Programming

A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most results for .E-convex functions and E-convex programming in [1] are not true by six counter examples.

Asymptotic Performance of Sparse Signal Detection Using Convex Programming Method

The detection of sparse signals against background noise is considered. Detecting signals of such kind is difficult since only a small portion of the signal carries information. Prior knowledge is usually assumed to ease detection. In this paper, we consider the general unknown and arbitrary sparse signal detection problem when no prior knowledge is available. Under a Ney- man-Pearson hypothesis-testing framework, a new detection scheme is proposed by combining a generalized likelihood ratio test （GLRT）-Iike test statistic and convex programming methods which directly exploit sparsity in an underdetermined system of linear equations. We characterize large sample behavior of the proposed method by analyzing its asymptotic performance. Specifically, we give the condition for the Chernoff-consistent detection which shows that the proposed method is very sensitive to the norm energy of the sparse signals. Both the false alam rate and the miss rate tend to zero at vanishing signal-to-noise ratio （SNR）, as long as the signal energy grows at least logarithmically with the problem dimension. Next we give a large deviation analysis to characterize the error exponent for the Neyman-Pearson detection. We derive the oracle error exponent assuming signal knowledge. Then we explicitly derive the error exponent of the proposed scheme and compare it with the oracle exponent. We complement our study with numerical experiments, showing that the proposed method performs in the vicinity of the likelihood ratio test （LRT） method in the finite sample scenario and the error probability degrades exponentially with the number of observations.

SEQUENTIAL CONVEX PROGRAMMING METHODS FOR SOLVING LARGE TOPOLOGY OPTIMIZATION PROBLEMS： IMPLEMENTATION AND COMPUTATIONAL RESULTS

In this paper, we describe a method to solve large-scale structural optimization problems by sequential convex programming （SCP）. A predictor-corrector interior point method is applied to solve the strictly convex subproblems. The SCP algorithm and the topology optimization approach are introduced. Especially, different strategies to solve certain linear systems of equations are analyzed. Numerical results are presented to show the efficiency of the proposed method for solving topology optimization problems and to compare different variants.

Relaxed inertial proximal Peaceman-Rachford splitting method for separable convex programming

The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman- Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman- Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.

A NEW DUAL PROBLEM FOR NONDIFFERENTIABLE CONVEX PROGRAMMING

This paper gives a new dual problem for nondifferentiable convex programming and provesthe properties of weak duality and strong duality and offers a necessary and sufficient condition ofstrong duality.

Block-Wise ADMM with a Relaxation Factor for Multiple-Block Convex Programming

It has been shown that the alternating direction method of multipliers(ADMM)is not necessarily convergent when it is directly extended to a multiple-block linearly constrained convex minimization model with an objective function that is in the sum of more than two functions without coupled variables.Recently,we pro-posed the block-wise ADMM,which was obtained by regrouping the variables and functions of such a model as two blocks and then applying the original ADMM in block-wise.This note is a further study on this topic with the purpose of showing that a well-known relaxation factor proposed by Fortin and Glowinski for iteratively updat-ing the Lagrangian multiplier of the originalADMMcan also be used in the block-wise ADMM.We thus propose the block-wise ADMM with Fortin and Glowinski’s relax-ation factor for the multiple-block convex minimization model.Like the block-wise ADMM,we also suggest further decomposing the resulting subproblems and regular-izing them by proximal terms to ensure the convergence.For the block-wise ADMM with Fortin and Glowinski's relaxation factor,its convergence and worst-case conver-gence rate measured by the iteration complexity in the ergodic sense are derived.

A POLYNOMIAL PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O（√nL） iteration complexity which is the best result for convex quadratic programming so far.

A Combined Homotopy Infeasible Interior-Point Method for Convex Nonlinear Programming

In this paper, on the basis of the logarithmic barrier function and KKT conditions, we propose a combined homotopy infeasible interior-point method （CHIIP） for convex nonlinear programming problems. For any convex nonlinear programming, without strict convexity for the logarithmic barrier function, we get different solutions of the convex programming in different cases by CHIIP method.

Fast First-Order Methods for Minimizing Convex Composite Functions

Two new versions of accelerated first-order methods for minimizing convex composite functions are proposed. In this paper, we first present an accelerated first-order method which chooses the step size 1/ Lk to be 1/ L0 at the beginning of each iteration and preserves the computational simplicity of the fast iterative shrinkage-thresholding algorithm. The first proposed algorithm is a non-monotone algorithm. To avoid this behavior, we present another accelerated monotone first-order method. The proposed two accelerated first-order methods are proved to have a better convergence rate for minimizing convex composite functions. Numerical results demonstrate the efficiency of the proposed two accelerated first-order methods.

Checking weak and strong optimality of the solution to interval convex quadratic program

In this paper,we investigate three canonical forms of interval convex quadratic pro-gramming problems.Necessary and suficient conditions for checking weak and strong optimality of given vector corresponding to various forms of feasible region,are established respectively.By using the concept of feasible direction,these conditions are formulated in the form of linear systems with both equations and inequalities.In addition,we provide two specific examples to illustrate the efficiency of the conditions.

Optimal deployment of swarm positions in cooperative interception of multiple UAV swarms

In order to prevent the attacker from breaking through the blockade of the interception,deploying multiple Unmanned Aerial Vehicle(UAV)swarms on the interception line is a new combat style.To solve the optimal deployment of swarm positions in the cooperative interception,an optimal deployment optimization model is presented by minimizing the penetration zones'area and the analytical expression of the optimal deployment positions is deduced.Firstly,from the view of the attackers breaking through the interception line,the situations of vertical penetration and oblique penetration are analyzed respectively,and the mathematical models of penetration zones are obtained under the condition of a single UAV swarm and multiple UAV swarms.Secondly,based on the optimization goal of minimizing the penetration area,the optimal deployment optimization model for swarm positions is proposed,and the analytical solution of the optimal deployment is solved by using the convex programming theory.Finally,the proposed optimal deployment is compared with the uniform deployment and random deployment to verify the validity of the theoretical analysis.

Solving the Binary Linear Programming Model in Polynomial Time

The paper presents a technique for solving the binary linear programming model in polynomial time. The general binary linear programming problem is transformed into a convex quadratic programming problem. The convex quadratic programming problem is then solved by interior point algorithms. This settles one of the open problems of whether P = NP or not. The worst case complexity of interior point algorithms for the convex quadratic problem is polynomial. It can also be shown that every liner integer problem can be converted into binary linear problem.

CONTINUUM TOPOLOGY OPTIMIZATION FOR MONOLITHIC COMPLIANT MECHANISMS OF MICRO-ACTUATORS

A multi-objective scheme for structural topology optimization of distributed compliant mechanisms of micro-actuators in MEMS condition is presented in this work, in which mechanical flexibility and structural stiffness are both considered as objective functions. The compliant micro-mechanism developed in this way can not only provide sufficient output work but also have sufficient rigidity to resist reaction forces and maintain its shape when holding the work-piece. A density filtering approach is also proposed to eliminate numerical instabilities such as checkerboards, mesh-dependency and one-node connected hinges occurring in resulting mechanisms. SIMP is used as the interpolation scheme to indicate the dependence of material modulus on element-regularized densities. The sequential convex programming method, such as the method of moving asymptotes （MMA）, is used to solve the optimization problem. The validation of the presented methodologies is demonstrated by a typical numerical example.

Projection type neural network and its convergence analysis

Projection type neural network for optimization problems has advantages over other networks for fewer parameters , low searching space dimension and simple structure. In this paper, by properly constructing a Lyapunov energy function, we have proven the global convergence of this network when being used to optimize a continuously differentiable convex function defined on a closed convex set. The result settles the extensive applicability of the network. Several numerical examples are given to verify the efficiency of the network.