本文選題：圓錐規(guī)劃　切入點(diǎn)：圓錐互補(bǔ)問(wèn)題　出處：《桂林電子科技大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：圓錐規(guī)劃和圓錐互補(bǔ)問(wèn)題是數(shù)學(xué)規(guī)劃領(lǐng)域的一個(gè)重要分支.圓錐規(guī)劃是在有限個(gè)圓錐笛卡爾積和仿射子空間的交集上求目標(biāo)函數(shù)的極小值或極大值問(wèn)題,而圓錐互補(bǔ)問(wèn)題是一類均衡優(yōu)化問(wèn)題.圓錐規(guī)劃和圓錐互補(bǔ)問(wèn)題被廣泛應(yīng)用于工程問(wèn)題,如多指手臂機(jī)器人的最優(yōu)抓力操縱問(wèn)題、接觸力優(yōu)化問(wèn)題和四足機(jī)器人力優(yōu)化問(wèn)題.但在標(biāo)準(zhǔn)內(nèi)積下,圓錐通常是非對(duì)稱錐,因此目前關(guān)于圓錐規(guī)劃和圓錐互補(bǔ)問(wèn)題的算法尚不多見.本文主要給出圓錐規(guī)劃和圓錐互補(bǔ)問(wèn)題的光滑牛頓法,并取得以下主要成果:1.基于一個(gè)光滑函數(shù)和圓錐與二階錐的代數(shù)關(guān)系,給出求解圓錐規(guī)劃的單調(diào)光滑牛頓法.運(yùn)用歐幾里得約當(dāng)代數(shù)理論,分析了算法的全局和局部二階收斂性.四足機(jī)器人的力優(yōu)化問(wèn)題和隨機(jī)生成的圓錐規(guī)劃問(wèn)題的數(shù)值結(jié)果表明新算法的有效性.2.給出求解圓錐規(guī)劃的非單調(diào)光滑牛頓法.為了提高新算法的收斂速度,在光滑牛頓法中引入非單調(diào)線搜索.在適當(dāng)?shù)募僭O(shè)下,證明了新算法是全局和局部二階收斂的.數(shù)值結(jié)果表明算法求解圓錐規(guī)劃問(wèn)題所需的計(jì)算時(shí)間和迭代次數(shù)都很少,且比較穩(wěn)定,從而說(shuō)明其有效性.3.基于一類帶參數(shù)的光滑函數(shù),將圓錐互補(bǔ)問(wèn)題轉(zhuǎn)化成非線性方程組,給出求解圓錐互補(bǔ)問(wèn)題的非單調(diào)光滑牛頓法.該算法中引入新的非單調(diào)線搜索,以取得更好的數(shù)值結(jié)果.在適定的條件下證明效益函數(shù)的強(qiáng)制性以及算法的全局和局部二階收斂性.通過(guò)數(shù)值結(jié)果驗(yàn)證算法的有效性.
[Abstract]:Cone programming and cone complementarity problems are important branches in the field of mathematical programming. Cone programming is a problem of finding the minimum or maximum value of objective functions on the intersection of finite cone Cartesian products and affine subspaces. Conical complementarity problem is a class of equilibrium optimization problems. Cone programming and cone complementarity problems are widely used in engineering problems, such as the optimal manipulating of multi-fingered manipulators. The contact force optimization problem and the quadruped robot force optimization problem. But under the standard inner product, the cone is usually asymmetric cone, Therefore, there are few algorithms for cone programming and cone complementarity problems. In this paper, the smooth Newton method for cone programming and cone complementarity problems is presented. On the basis of a smooth function and the algebraic relation between a cone and a second order cone, a monotone smooth Newton method for solving cone programming is given. The Euclidean approximate contemporary number theory is used. The global and local second order convergence of the algorithm is analyzed. The numerical results of the force optimization problem for quadruped robot and the randomly generated cone programming problem show the effectiveness of the new algorithm. 2. The nonmonotone smooth cattle for solving the cone programming is given. In order to improve the convergence rate of the new algorithm, Non-monotone line search is introduced into the smooth Newton method. Under appropriate assumptions, it is proved that the new algorithm is global and locally second-order convergent. The numerical results show that the computational time and the number of iterations required for the algorithm to solve the conical programming problem are few. Based on a class of smooth functions with parameters, the cone complementarity problem is transformed into nonlinear equations. A nonmonotone smooth Newton method for solving conical complementarity problems is presented, in which a new nonmonotone line search is introduced. In order to obtain better numerical results, the mandatory benefit function and the global and local second-order convergence of the algorithm are proved under suitable conditions. The validity of the algorithm is verified by numerical results.
【學(xué)位授予單位】：桂林電子科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O221

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本文編號(hào)：1598817