一类积分型全局最优性条件及其应用研究
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摘要
对全局最优的刻画一直是数学规划领域最核心的研究内容之一。已有研究结果多以最优性条件的形式给出,依据所使用的运算工具,大致可划分为微分型与积分型两类。微分型最优性条件以导数/广义导数运算为基础,且多为局部最优条件,目前已有的绝大多数研究均属此类。积分型最优性条件则以积分运算为基础构建,考虑到积分运算的‘凸化’效果,该类型最优性条件能够较好的刻画问题的全局最优解与最优值,但已有研究相对较少。
本文研究一类积分型全局最优性条件,包含分别对全局最优值与全局最优解的积分刻画。对此积分型全局最优性条件及其性质、数值计算及在若干优化问题中的应用进行了研究,具体内容如下：
1.第二章为积分型全局最优性条件及其性质与数值实现的研究。首先追溯了积分型全局最优性条件的历史,指出该条件中关于最优值的表述源于Laplace方法,而对最优解的表述最早由Pincus提出。然后对此最优条件进行研究,一方而,指出若可行域由紧致集松弛为Borel集(不一定有界),最优值积分型全局最优性条件依然成立；若优化问题的全局最优解不唯一为有限多个时,Pincus所指出的最优解积分型全局最优性条件将不再成立,此时得到的是这有限多个全局最优解的某个凸组合。另一方面,基于最优解的积分型全局最优性条件,给出了求解全局优化的一个概念性算法,并结合蒙特卡洛方法,给出算法的数值实现。抽取典型全局优化(包括无约束/箱式约束、约束及非光滑)考题进行了数值验证。计算结果表明该算法可以得到全部考题的全局最优解与最优值,且具有无需初始点、对函数性态要求较低且可保证解的全局最优等特点。最后,对可微优化问题,研究了积分型全局最优性条件与传统微分型最优性条件的联系。
2.第三章为积分型全局最优性条件在极大极小问题中的应用研究。一方面,研究离散测度下的积分型全局最优性条件在有限极大极小问题中的应用,得到的恰为求解有限极大极小问题的凝聚函数法(又称为熵函数法),指出凝聚函数富含极大值函数的高阶信息。特别的,对于凸问题,(叉熵)凝聚函数的一阶导数,在光滑参数趋向于无穷大时,为极大值函数的某个次梯度。另一方面,研究Lebesgue测度下的积分型全局最优性条件在半无限极大极小问题中的应用,得到半无限极大极小问题的一类积分型光滑逼近。对由此得到的积分型光滑函数的性质进行了研究,特别的,澄清了已有一L作中对此光滑函数关于光滑参数单调性的误解。而对此光滑函数的导数计算表明,该函数富含极大值函数的统计信息,并通过对其共轭函数—熵函数的Legendre级数展开对此现象进行了初步阐释。
3.第四章为积分型全局最优性条件在变分不等式中的应用研究。基于最优值的积分型全局最优性条件,得到了变分不等式的一个新的光滑gap函数,利用此光滑gap函数可将变分不等式转化为一个等价的光滑优化问题。并结合最优解的积分型全局最优性条件,设计了基于此光滑gap函数的下降类算法。详细讨论了光滑gap函数的性质,其中,稳定点特性表明,在一定条件下,由任意初始点得到的等价优化问题的稳定点皆为全局最优解,即为变分不等式问题的解。基于光滑gap函数的下降方向,分别结合精确线搜索与Armijo非精确线搜索给出了求解变分不等式问题的下降类算法。该类算法无需对函数的Hessian阵进行计算,从而节省计算量与内存空间,并证明了其全局收敛性。针对箱式约束变分不等式问题,给出了光滑gap函数、gap函数下降方向及下降类算法的显式表达,并抽取典型考题对下降类算法进行数值验证。
4.第五章为积分型全局最优性条件在锥约束优化中的应用研究。借助于值函数及其共轭函数,应用最优值积分型全局最优性条件,得到凸锥约束下的凸规划问题及其对偶问题的对数障碍函数,从而对对数障碍函数的由来给出一个有趣的阐释。进一步地,基于对对数障碍函数推导过程的分析,给出一类利用积分型全局最优性条件构造对偶问题的新方法,若值函数为凹的,则此对偶体系的强对偶条件成立,即原问题与该方法所构造对偶问题的最优值相等。尤其对于利用共轭运算构造对偶问题而较难计算的问题,该方法经运算可得到显式的对偶问题。
The characterization of global optimality, which is studied as the optimality condition, is one of the most important and corn issues in the field of mathematical programming. From the viewpoint of mathematical operation, the formulation of optimality condition can be divided into two types, i.e. the differential one and the integral one. The differential type of optimality condition, which is also known as the gradient-based one, has been studied much more extensively and profoundly during the past50years, although it just describes a local optimal characteristic. Contrast to the differential one, the integral type of optimality condition is often viewed as the global optimality condition since the integral operation may have some convexity effects on the approximated functions. However, the studies of integral optimality conditions are still relatively few until now.
In this dissertation, we mainly study a class of integral global optimality condition, which is composed of the integral characteristics of global optimal value and the integral characteristics of global optimal solution. We investigate the theoretical properties, computational aspects and some applications of the IGOC.
The main results of this thesis are as follows:
1. Some basic theoretical properties and computational aspects of IGOC are studied in chapter2. Firstly, we review the literature and point out that the IGOC of optimal value probably goes back to Laplace' method and the IGOC of optimal solution was discovered by Pincus. Based on this, we point out that if the optimization problem has finite global optima, Pincus' condition will result in a convex combination of the global optimal vectors. Secondly, we study the computational aspects of IGOC and propose a conceptual algorithm for global optimization problems based on the IGOC of optimal solution. In the proposed algorithm, Monte-Carlo method is employed to compute the integration in the formula of IGOC. Numerical results on a set of famous optimization test problems (including unconstrained/box constrained, constrained and nons-mooth problems) show that our algorithm owns some good features, such as does not require the initial point, does not require the differentiability of functions and always can obtains the global optimal solution. At last, we also reveal the relationships between the IGOC of optimal value and the classical gradient-based optimality conditions.
2. We analyze the application of1GOC in min-max problem in chapter3. On one hand, we discuss the IGOC with discrete measure and finite min-max problem. In this situation, we obtain the aggregate function method (also called entropic method), which is a mature method for finite min-max problem. The aggregate function in entropic method contains high-order information of max-function. Especially, the first-order gradient of aggregate function is just one sub-gradient of max-function with convex components, when the smoothing parameter tends to infinity. On the other hand, we discuss the IGOC with Lebesgue measure and semi-infinite min-max problem. We obtain one integral smooth approximation of semi-infinite min-max problem and discuss the detailed properties of the integral smoothing function. Especially, we clarify the mistake of monotonicity of this integral smoothing function with the smoothing parameter. Furthermore, we propose that this integral smoothing function contains high-order information of max-function and give an explanation of this by the Legendre series expansion of entropy function.
3. In chapter4, a new gap function which is a smooth approximation of Auslender's gap function is obtained by the using of IGOC of optimal value in variational inequality problem (VIP). And VIP can be formulated as an equivalent differentiable optimization problem. We discuss properties of the new smooth gap function. Especially, under appropriate assumptions on VIP, any stationary point of the optimization problem is a global optimal solution, and hence solves the VIP. Based on the IGOC of optimal solution, we discuss descent methods for solving the equivalent optimization problem. Two algorithms that uses the descent direction with exact line search rule and with Armijo-type steplength rule are proposed. The global convergence results of these algorithms are also established. At last, the analytic expressions of smoothing gap function、descent direction and the descent algorithms for box VIP are proposed. Some test problems are calculated to confirm the descent algorithm.
4. In chapter5, the log-barrier function and a systematic way to obtain a dual problem is obtained by the using of IGOC of optimal value in cone-constrained optimization. With value function and its conjugate function, we obtain the log-barrier function for cone-constrained optimization. Furthermore, the application technique of IGOC with value function and its conjugate function provides a systematic duality method in cases when dual problem can not be obtained explicitly from the conjugation.
本文研究一类积分型全局最优性条件,包含分别对全局最优值与全局最优解的积分刻画。对此积分型全局最优性条件及其性质、数值计算及在若干优化问题中的应用进行了研究,具体内容如下：
1.第二章为积分型全局最优性条件及其性质与数值实现的研究。首先追溯了积分型全局最优性条件的历史,指出该条件中关于最优值的表述源于Laplace方法,而对最优解的表述最早由Pincus提出。然后对此最优条件进行研究,一方而,指出若可行域由紧致集松弛为Borel集(不一定有界),最优值积分型全局最优性条件依然成立；若优化问题的全局最优解不唯一为有限多个时,Pincus所指出的最优解积分型全局最优性条件将不再成立,此时得到的是这有限多个全局最优解的某个凸组合。另一方面,基于最优解的积分型全局最优性条件,给出了求解全局优化的一个概念性算法,并结合蒙特卡洛方法,给出算法的数值实现。抽取典型全局优化(包括无约束/箱式约束、约束及非光滑)考题进行了数值验证。计算结果表明该算法可以得到全部考题的全局最优解与最优值,且具有无需初始点、对函数性态要求较低且可保证解的全局最优等特点。最后,对可微优化问题,研究了积分型全局最优性条件与传统微分型最优性条件的联系。
2.第三章为积分型全局最优性条件在极大极小问题中的应用研究。一方面,研究离散测度下的积分型全局最优性条件在有限极大极小问题中的应用,得到的恰为求解有限极大极小问题的凝聚函数法(又称为熵函数法),指出凝聚函数富含极大值函数的高阶信息。特别的,对于凸问题,(叉熵)凝聚函数的一阶导数,在光滑参数趋向于无穷大时,为极大值函数的某个次梯度。另一方面,研究Lebesgue测度下的积分型全局最优性条件在半无限极大极小问题中的应用,得到半无限极大极小问题的一类积分型光滑逼近。对由此得到的积分型光滑函数的性质进行了研究,特别的,澄清了已有一L作中对此光滑函数关于光滑参数单调性的误解。而对此光滑函数的导数计算表明,该函数富含极大值函数的统计信息,并通过对其共轭函数—熵函数的Legendre级数展开对此现象进行了初步阐释。
3.第四章为积分型全局最优性条件在变分不等式中的应用研究。基于最优值的积分型全局最优性条件,得到了变分不等式的一个新的光滑gap函数,利用此光滑gap函数可将变分不等式转化为一个等价的光滑优化问题。并结合最优解的积分型全局最优性条件,设计了基于此光滑gap函数的下降类算法。详细讨论了光滑gap函数的性质,其中,稳定点特性表明,在一定条件下,由任意初始点得到的等价优化问题的稳定点皆为全局最优解,即为变分不等式问题的解。基于光滑gap函数的下降方向,分别结合精确线搜索与Armijo非精确线搜索给出了求解变分不等式问题的下降类算法。该类算法无需对函数的Hessian阵进行计算,从而节省计算量与内存空间,并证明了其全局收敛性。针对箱式约束变分不等式问题,给出了光滑gap函数、gap函数下降方向及下降类算法的显式表达,并抽取典型考题对下降类算法进行数值验证。
4.第五章为积分型全局最优性条件在锥约束优化中的应用研究。借助于值函数及其共轭函数,应用最优值积分型全局最优性条件,得到凸锥约束下的凸规划问题及其对偶问题的对数障碍函数,从而对对数障碍函数的由来给出一个有趣的阐释。进一步地,基于对对数障碍函数推导过程的分析,给出一类利用积分型全局最优性条件构造对偶问题的新方法,若值函数为凹的,则此对偶体系的强对偶条件成立,即原问题与该方法所构造对偶问题的最优值相等。尤其对于利用共轭运算构造对偶问题而较难计算的问题,该方法经运算可得到显式的对偶问题。
The characterization of global optimality, which is studied as the optimality condition, is one of the most important and corn issues in the field of mathematical programming. From the viewpoint of mathematical operation, the formulation of optimality condition can be divided into two types, i.e. the differential one and the integral one. The differential type of optimality condition, which is also known as the gradient-based one, has been studied much more extensively and profoundly during the past50years, although it just describes a local optimal characteristic. Contrast to the differential one, the integral type of optimality condition is often viewed as the global optimality condition since the integral operation may have some convexity effects on the approximated functions. However, the studies of integral optimality conditions are still relatively few until now.
In this dissertation, we mainly study a class of integral global optimality condition, which is composed of the integral characteristics of global optimal value and the integral characteristics of global optimal solution. We investigate the theoretical properties, computational aspects and some applications of the IGOC.
The main results of this thesis are as follows:
1. Some basic theoretical properties and computational aspects of IGOC are studied in chapter2. Firstly, we review the literature and point out that the IGOC of optimal value probably goes back to Laplace' method and the IGOC of optimal solution was discovered by Pincus. Based on this, we point out that if the optimization problem has finite global optima, Pincus' condition will result in a convex combination of the global optimal vectors. Secondly, we study the computational aspects of IGOC and propose a conceptual algorithm for global optimization problems based on the IGOC of optimal solution. In the proposed algorithm, Monte-Carlo method is employed to compute the integration in the formula of IGOC. Numerical results on a set of famous optimization test problems (including unconstrained/box constrained, constrained and nons-mooth problems) show that our algorithm owns some good features, such as does not require the initial point, does not require the differentiability of functions and always can obtains the global optimal solution. At last, we also reveal the relationships between the IGOC of optimal value and the classical gradient-based optimality conditions.
2. We analyze the application of1GOC in min-max problem in chapter3. On one hand, we discuss the IGOC with discrete measure and finite min-max problem. In this situation, we obtain the aggregate function method (also called entropic method), which is a mature method for finite min-max problem. The aggregate function in entropic method contains high-order information of max-function. Especially, the first-order gradient of aggregate function is just one sub-gradient of max-function with convex components, when the smoothing parameter tends to infinity. On the other hand, we discuss the IGOC with Lebesgue measure and semi-infinite min-max problem. We obtain one integral smooth approximation of semi-infinite min-max problem and discuss the detailed properties of the integral smoothing function. Especially, we clarify the mistake of monotonicity of this integral smoothing function with the smoothing parameter. Furthermore, we propose that this integral smoothing function contains high-order information of max-function and give an explanation of this by the Legendre series expansion of entropy function.
3. In chapter4, a new gap function which is a smooth approximation of Auslender's gap function is obtained by the using of IGOC of optimal value in variational inequality problem (VIP). And VIP can be formulated as an equivalent differentiable optimization problem. We discuss properties of the new smooth gap function. Especially, under appropriate assumptions on VIP, any stationary point of the optimization problem is a global optimal solution, and hence solves the VIP. Based on the IGOC of optimal solution, we discuss descent methods for solving the equivalent optimization problem. Two algorithms that uses the descent direction with exact line search rule and with Armijo-type steplength rule are proposed. The global convergence results of these algorithms are also established. At last, the analytic expressions of smoothing gap function、descent direction and the descent algorithms for box VIP are proposed. Some test problems are calculated to confirm the descent algorithm.
4. In chapter5, the log-barrier function and a systematic way to obtain a dual problem is obtained by the using of IGOC of optimal value in cone-constrained optimization. With value function and its conjugate function, we obtain the log-barrier function for cone-constrained optimization. Furthermore, the application technique of IGOC with value function and its conjugate function provides a systematic duality method in cases when dual problem can not be obtained explicitly from the conjugation.
引文
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