等式与界约束非线性优化信赖域算法的全局收敛

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｏｐｔｉｍｉｚａｔｉｏｎｐｒｏｂｌｅｍ :ｍｉｎｉｍｉｚｅｆ(ｘ)ｓｕｂｊｅｃｔｔｏＣ(ｘ) =0 ,　ａ≤ｘ≤ｂ ,( 1 .1 )ｗｈｅｒｅｆ(ｘ) :Ｒｎ→Ｒ ,Ｃ(ｘ) =(ｃ1(ｘ) ,ｃ2 (ｘ) ,...,ｃｍ(ｘ) ) Ｔ:Ｒｎ→Ｒｍ ａｒｅｔｗｉｃｅｃｏｎｔｉｎｕｏｕｓｌｙｄｉｆｆｅｒｅｎｔｉａｂｌｅ,ｍ≤ｎ ,ａ ,ｂ∈Ｒｎ.Ｔｒｕｓｔｒｅｇｉｏｎａｌｇｏｒｉｔｈｍｓａｒｅｖｅｒｙｅｆｆｅｃｔｉｖｅｆｏｒｓｏｌｖｉｎｇｎｏｎｌｉｎｅａｒｏｐｔｉｍｉ…     

Generalized Higher-Order Algebraic Differential Equations with Admissible Algebroid Solutions

§ 1.ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓ　ＩｎｔｈｉｓｐａｐｅｒｗｅｕｓｅｔｈｅｓｔａｎｄａｒｄｎｏｔａｔｉｏｎｓｏｆｔｈｅＮｅｖａｎｌｉｎｎａｔｈｅｏｒｙｏｆｍｅｒｏｍｏｒｐｈｉｃｆｕｎｃｔｉｏｎｓｏｒａｌｇｅｂｒｏｉｄｆｕｎｃｔｉｏｎｓ.Ｆｏｒｒｅｆｅｒｅｎｃｅｓｅｅｅ.ｇ.[1 ] .　Ｃｏｎｓｉｄｅｒｔｈｅｇｅｎｅｒａｌｉｚｅｄｈｉｇｈｅｒ ｏｒｄｅｒａｌｇｅｂｒａｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎΩ1(ｚ,ｗ)Ω2 (ｚ,ｗ) =Ｐ(ｚ,ｗ)Ｑ(ｚ,ｗ) , ( 1 )ｗｈｅｒｅΩ1(ｚ,ｗ) =∑…     

Young Measure Solutions of a Class of Singular Diffusion Equations

Ｃｏｎｓｉｄｅｒｔｈｅｆｉｒｓｔｉｎｉｔｉａｌ ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ ｕ ｔ=ｄｉｖ ｑ( ｕ) ,　 (ｘ,ｔ) ∈ＱＴ,(1 )ｕ(ｘ,ｔ) =0 ,　 (ｘ,ｔ) ∈ Ω× (0 ,Ｔ) ,(2 )ｕ(ｘ,0 ) =ｕ0 (ｘ) ,　ｘ∈Ω ,(3 )ｗｈｅｒｅΩｉｓａｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮｗｉｔｈｓｍｏｏｔｈｂｏｕｎｄａｒｙ Ω ,ＱＴ=Ω× (0 ,Ｔ) , ｑ = φ ,φ∈Ｃ1(ＲＮ) ,ａｎｄ φ , ｑｓａｔｉｓｆｙｔｈｅｓｔｒｕｃｔｕｒｅｃｏｎｄｉｔｉｏｎｓ(λ｜ξ｜1+δ-1 ) +≤ φ(ξ) ≤Λ｜ξ｜1+δ+ 1 ,　 ξ∈ＲＮ,(4 )｜ ｑ(ξ) …     

Remarks on a Mathematical Model from the Theory of Optimal Investment

ＩｎＣｈａｐｔｅｒ 4ｏｆ[1 ] ,ｆｒｏｍｔｈｅｔｈｅｏｒｙｏｆｏｐｔｉｍａｌｉｎｖｅｓｔｍｅｎｔｉｎｍａｔｈｅｍａｔｉｃａｌｆｉｎａｎｃｅ ,ｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｖａｌｕｅｐｒｏｂｌｅｍｆｏｒａｐａｒａｂｏｌｉｃＭｏｎｇｅ Ａｍｐ埁ｒｅｅｑｕａｔｉｏｎｗａｓｄｅｒｉｖｅｄ :ＶｓＶｙｙ+ｒｙＶｙＶｙｙ-ｂ-ｒσ Ｖ2 ｙ =0 ,　 (ｙ ,ｓ) ∈Ｒ× [0 ,Ｔ) ,Ｖｙｙ <0 ,Ｖ(ｙ ,Ｔ) =ｇ(ｙ) ,　ｙ∈Ｒ( 1 )ｗｈｅｒｅＶ =Ｖ(ｙ,ｓ)ｉｓｔｈｅｕｎｋｎｏｗｎｆｕｎｃｔｉｏｎ ,ｒ,ｂ ,σａｒｅｇｉｖｅｎｃｏｎ…     

EXISTENCE TIME FOR THE SEMILINEAR SCHR?DINGER EQUATION

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ ,ｗｅｗｉｌｌｃｏｎｓｉｄｅｒｔｈｅｓｅｍｉｌｉｎｅａｒＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｉｎｏｎｅｓｐａｃｅｄｉｍｅｎｓｉｏｎｏｆｔｈｅｔｙｐｅｕｔ-ｉｕｘｘ =Ｆ(ｕ) .　 (ｘ ,ｔ)∈Ｒ×Ｒ+,( 1 )ｕ(ｘ ,0 ) =ｕ0 ,　ｘ ∈Ｒ ,( 2 )ｗｈｅｒｅｕ =ｕ(ｘ ,ｔ)ｉｓｃｏｍｐｌｅｘ ｖａｌｕｅｄｆｕｎｃｔｉｏｎ ,ａｎｄＦｉｓａｓｍｏｏｔｈｆｕｎｃｔｉｏｎｏｆｕｓｕｃｈｔｈａｔ｜Ｆ(ｕ) ｜ =Ｏ( ｜ｕ｜α+1)ｆｏｒ |ｕ|ｓｕｆｆｉｃｉｅｎｔｌｙｓｍａｌｌａｎｄ…     

Heisenberg群上四阶非线性次椭圆方程的可解性

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｔｈｅｆｏｕｒｔｈｏｒｄｅｒｓｅｍｉｌｉｎｅａｒｓｕｂｅｌｌｉｐｔｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍΔ2 Ｈｕ +ｃΔＨｕ =ｆ( (ｚ ,ｔ) ,ｕ)　ｉｎＤ ,ｕ｜Ｄ =ΔＨｕ｜Ｄ =0 ,( 1 .1 )ｗｈｅｒｅＤｉｓａｂｏｕｎｄｅｄｏｐｅｎｓｕｂｓｅｔｏｆｔｈｅＨｅｉｓｅｎｂｅｒｇｇｒｏｕｐＨｎａｎｄΔＨｉｓｔｈｅｓｕｂｅｌｌｉｐｔｉｃＬａｐｌａ ｃｉａｎｏｎＨｎ.ＷｅｒｅｃａｌｌｔｈａｔＨｎｉｓｔｈｅＬｉｅｇｒｏｕｐｗｈｏｓｅｕｎｄｅｒｌｙｉｎｇｍａｎｉ…     

Several Existence Theorems for Positive Radial Solutions to a Semilinear Elliptic BVP

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｓｉｎｃｅｓｅｃｏｎｄｏｒｄｅｒｅｌｌｉｐｔｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｈａｖｅｗｉｄｅａｐｐｌｉｃａｔｉｏｎｓｉｎｐｈｙｓｉｃｓ ,ｔｈｅｓｔｕｄｙｏｆｐｏｓｉｔｉｖｅｒａｄｉａｌｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｓｅｍｉｌｉｎｅａｒｐｒｏｂｌｅｍ　 (Ｐ) Δｕ(Ｘ) +ｇ( |Ｘ|)ｆ(ｕ(Ｘ) ) =0 ,　Ｒ1<|Ｘ|<Ｒ2 ,ｕ(Ｘ) =0 ,　 |Ｘ|=Ｒ1ｏｒ |Ｘ|=Ｒ2(ｗｈｅｒｅＲ1>0 ,Ｘ∈Ｒｎ,ｎ≥ 2 )ｈａｓｂｅｅｎｍａｄｅｆｏｒｒｅｃｅｎｔ 2 0ｙｅａｒｓ ;ｓｅｅ [1— 5]ｆｏｒｒ…     

上期课外练习解答

高一年级1.设 ｆ(ｔ) =ｔ3 +2 0 0 3ｔ,易证 ｆ(ｔ)在Ｒ上是奇函数且递增函数 ,由题意可知 :ｆ(ｘ - 1) =- 1,　ｆ(ｙ - 1) =1.即　ｆ(ｘ - 1) =-ｆ( ｙ - 1) =ｆ( 1-ｙ) .∴　ｘ - 1=1-ｙ ,故ｘ +ｙ =2 .2 .由条件知 :ｓｉｎαｃｏｓβ2 0 0 2 ,ｓｉｎβｃｏｓα2 0 0 2 中必有一个不大于 1,一个不小于 1.不妨设　 ｓｉｎαｃｏｓβ2 0 0 2 ≤ 1,　 ｓｉｎβｃｏｓα2 0 0 2 ≥ 1.∵　α ,β∈ ( 0 ,π2 ) ,又ｙ=ｓｉｎｘ在 ( 0 ,π2 )上递增 .∴　ｓｉｎα≤ｃｏｓβ且ｓｉｎβ≥ｃｏｓα .∴　ｓｉｎα≤ｓｉｎ( π2 - β)且ｓｉｎβ≥ｓ…     

用“若a+b=0,则ab≤0”解竞赛题

如果两实数ａ ,ｂ满足ａ +ｂ =0 ,则ａｂ≤0 .应用这个结论解答一些竞赛题十分简捷 .现举例说明 .例 1　ｘ ,ｙ ,ｚ均为实数 ,解方程组ｘ + ｙ =2ｘｙ -ｚ2 =1①②(1987年上海市初中数学竞赛 )解　由①得　 (ｘ -1) + (ｙ -1) =0 .∴ (ｘ -1) (ｙ -1) =ｘｙ-(ｘ + ｙ) + 1≤0 ③①、②代入③得　 (ｘ -1) (ｙ -1) =ｚ2 ≤ 0 ,∴　ｚ =0 ,　ｘ -1=ｙ -1=0 .故方程组的解是　ｘ =1,ｙ =1,ｚ =0 .例 2　已知实数ａ ,ｂ ,ｃ满足ａ +ｂ +ｃ =0 ,ａｂｃ=8.求ｃ的取值范围 .(第一届“希望杯”初二数学竞赛 )解　由已知 (ａ + 12 ｃ) + (ｂ + 12 ｃ)…     

一道全国联赛题的别解

1999年全国高中数学联合竞赛加试试题第二题是 :给定实数ａ ,ｂ ,ｃ .已知复数ｚ1 ,ｚ2 ,ｚ3满足 :|ｚ1 |=|ｚ2 |=|ｚ3|=1.ｚ1 ｚ2 ｚ2ｚ3 ｚ3ｚ1=1.求 |ａｚ1 ｂｚ2 ｃｚ3|的值 .命题委员会提供的“参考答案”用到了关于复数的欧拉公式ｅｉθ=ｃｏｓθ ｉｓｉｎθ .下面我们给出此题的一种简便的解法 .解　令ｚ1 =ｃｏｓθ1 ｉｓｉｎθ1 ,ｚ2 =ｃｏｓθ2 ｉｓｉｎθ2 ,ｚ3=ｃｏｓθ3 ｉｓｉｎθ3,则ｚ1 ｚ2 ｚ2ｚ3 ｚ3ｚ1=ｃｏｓ(θ1 -θ2 ) ｃｏｓ(θ2 -θ3) ｃｏｓ(θ3-θ1 ) [ｓｉｎ(θ1 -θ2 ) ] ｓｉｎ(θ2-θ3) ｓｉｎ(θ3…     

Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.     

Convexifiability of continuous and discrete nonnegative quadratic programs for gap-free duality

In this paper we show that a convexifiability property of nonconvex quadratic programs with nonnegative variables and quadratic constraints guarantees zero duality gap between the quadratic programs and their semi-Lagrangian duals. More importantly, we establish that this convexifiability is hidden in classes of nonnegative homogeneous quadratic programs and discrete quadratic programs, such as mixed integer quadratic programs, revealing zero duality gaps. As an application, we prove that robust counterparts of uncertain mixed integer quadratic programs with objective data uncertainty enjoy zero duality gaps under suitable conditions. Various sufficient conditions for convexifiability are also given.     

Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.     

On zero duality gap in nonconvex quadratic programming problems

We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap.     

均衡约束数学规划问题的一类广义Mond-Weir型对偶理论

针对均衡约束数学规划模型难以满足约束规范及难于求解的问题,基于Mond和Weir提出的标准非线性规划的对偶形式,利用其S稳定性,建立了均衡约束数学规划问题的一类广义Mond-Weir型对偶,从而为求解均衡约束优化问题提供了一种新的方法.在Hanson-Mond广义凸性条件下,利用次线性函数,分别提出了弱对偶性、强对偶性和严格逆对偶性定理,并给出了相应证明.该对偶化方法的推广为研究均衡约束数学规划问题的解提供了理论依据.     

Reduction of indefinite quadratic programs to bilinear programs

Indefinite quadratic programs with quadratic constraints can be reduced to bilinear programs with bilinear constraints by duplication of variables. Such reductions are studied in which: (i) the number of additional variables is minimum or (ii) the number of complicating variables, i.e., variables to be fixed in order to obtain a linear program, in the resulting bilinear program is minimum. These two problems are shown to be equivalent to a maximum bipartite subgraph and a maximum stable set problem respectively in a graph associated with the quadratic program. Non-polynomial but practically efficient algorithms for both reductions are thus obtaine.d Reduction of more general global optimization problems than quadratic programs to bilinear programs is also briefly discussed.     

DC approach to weakly convex optimization and nonconvex quadratic optimization problems

Motivated by weakly convex optimization and quadratic optimization problems, we first show that there is no duality gap between a difference of convex (DC) program over DC constraints and its associated dual problem. We then provide certificates of global optimality for a class of nonconvex optimization problems. As an application, we derive characterizations of robust solutions for uncertain general nonconvex quadratic optimization problems over nonconvex quadratic constraints.     

Support vector machine classifiers with uncertain knowledge sets via robust optimization

In this article we study support vector machine (SVM) classifiers in the face of uncertain knowledge sets and show how data uncertainty in knowledge sets can be treated in SVM classification by employing robust optimization. We present knowledge-based SVM classifiers with uncertain knowledge sets using convex quadratic optimization duality. We show that the knowledge-based SVM, where prior knowledge is in the form of uncertain linear constraints, results in an uncertain convex optimization problem with a set containment constraint. Using a new extension of Farkas' lemma, we reformulate the robust counterpart of the uncertain convex optimization problem in the case of interval uncertainty as a convex quadratic optimization problem. We then reformulate the resulting convex optimization problems as a simple quadratic optimization problem with non-negativity constraints using the Lagrange duality. We obtain the solution of the converted problem by a fixed point iterative algorithm and establish the convergence of the algorithm. We finally present some preliminary results of our computational experiments of the method.     

Minimizing a Sum of Norms Subject to Linear Equality Constraints

Numerical analysis of a class of nonlinear duality problems is presented. One side of the duality is to minimize a sum of Euclidean norms subject to linear equality constraints (the constrained MSN problem). The other side is to maximize a linear objective function subject to homogeneous linear equality constraints and quadratic inequalities. Large sparse problems of this form result from the discretization of infinite dimensional duality problems in plastic collapse analysis.The solution method is based on the l ₁ penalty function approach to the constrained MSN problem. This can be formulated as an unconstrained MSN problem for which the first author has recently published an efficient Newton barrier method, and for which new methods are still being developed.Numerical results are presented for plastic collapse problems with up to 180000 variables, 90000 terms in the sum of norms and 90000 linear constraints. The obtained accuracy is of order 10^-8 measured in feasibility and duality gap.     

New strong duality results for convex programs with separable constraints

It is known that convex programming problems with separable inequality constraints do not have duality gaps. However, strong duality may fail for these programs because the dual programs may not attain their maximum. In this paper, we establish conditions characterizing strong duality for convex programs with separable constraints. We also obtain a sub-differential formula characterizing strong duality for convex programs with separable constraints whenever the primal problems attain their minimum. Examples are given to illustrate our results.