In this work, we show the consistency of an approach for solving robust optimization problems using sequences of sub-problems generated by ergodic measure preserving transformations. The main result of this paper is t...
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In this work, we show the consistency of an approach for solving robust optimization problems using sequences of sub-problems generated by ergodic measure preserving transformations. The main result of this paper is that the minimizers and the optimal value of the sub-problems converge, in some sense, to the minimizers and the optimal value of the initial problem, respectively. Our result particularly implies the consistency of the scenario approach for nonconvex optimization problems. Finally, we show that our method can also be used to solve infinite programming problems.
An infinite programming problem consists in minimizing a functional defined on a real Banach space under an infinite number of constraints. The main purpose of this article is to provide sufficient conditions of optim...
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An infinite programming problem consists in minimizing a functional defined on a real Banach space under an infinite number of constraints. The main purpose of this article is to provide sufficient conditions of optimality under generalized convexity assumptions. Such conditions are necessarily satisfied when the problem possesses the property that every stationary point is a global minimizer.
This paper provides necessary and sufficient optimality conditions for abstract-constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geom...
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This paper provides necessary and sufficient optimality conditions for abstract-constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections.
Contact-implicit trajectory optimization (CITO) is an effective method to plan complex trajectories for various contact-rich systems including manipulation and locomotion. CITO formulates a mathematical program with c...
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Contact-implicit trajectory optimization (CITO) is an effective method to plan complex trajectories for various contact-rich systems including manipulation and locomotion. CITO formulates a mathematical program with complementarity constraints (MPCC) that enforces that contact forces must be zero when points are not in contact. However, MPCC solve times increase steeply with the number of allowable points of contact, which limits CITO's applicability to problems in which only a few, simple geometries are allowed us to make contact. This article introduces simultaneous trajectory optimization and contact selection (STOCS), as an extension of CITO that overcomes this limitation. The innovation of STOCS is to identify salient contact points and times inside the iterative trajectory optimization process. This effectively reduces the number of variables and constraints in each MPCC invocation. The STOCS framework, instantiated with key contact identification subroutines, renders the optimization of manipulation trajectories computationally tractable even for high-fidelity geometries consisting of tens of thousands of vertices.
Convex semi-definite semi-infinite programming problems (SDSIP) represent a special class of distributionally robust optimization (DRO) problems with a wide range of applications in engineering and economics. In this ...
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Convex semi-definite semi-infinite programming problems (SDSIP) represent a special class of distributionally robust optimization (DRO) problems with a wide range of applications in engineering and economics. In this paper, we propose a modified exchange algorithm for convex SDSIP that arises from DRO with matrix moment constraints. We first explore the convergence results of the modified exchange algorithm and perform the efficiency analysis based on a set of benchmark tests. In addition, we apply the SDSIP framework to investigate an optimized certainty bound risk with an ambiguity uncertainty set and implement the algorithm to solve a practical risk minimization problem in portfolio selection. The empirical results show both the efficiency of the algorithm and the robustness of the risk measure.
This paper presents an exact formula for computing the normal cones of the constraint set mapping including the Clarke normal cone and the Mordukhovich normal cone in infinite programming under the extended Mangasaria...
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This paper presents an exact formula for computing the normal cones of the constraint set mapping including the Clarke normal cone and the Mordukhovich normal cone in infinite programming under the extended Mangasarian-Fromovitz constraint qualification condition. Then, we derive an upper estimate as well as an exact formula for the limiting subdifferential of the marginal/optimal value function in a general Banach space setting.
An exact estimate on the modulus of metric regularity for linear systems is given. By applying the estimate, we obtain explicit forms of the modulus for linear conical systems and differentiable nonlinear systems on t...
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An exact estimate on the modulus of metric regularity for linear systems is given. By applying the estimate, we obtain explicit forms of the modulus for linear conical systems and differentiable nonlinear systems on the space of continuous functions.
As data become heterogeneous, multiple kernel learning methods may help to classify them. To overcome the drawback lying in its (multiple) finite choice, we propose a novel method of 'infinite' kernel combinat...
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As data become heterogeneous, multiple kernel learning methods may help to classify them. To overcome the drawback lying in its (multiple) finite choice, we propose a novel method of 'infinite' kernel combinations for learning problems with the help of infinite and semi-infinite optimizations. Looking at all the infinitesimally fine convex combinations of the kernels from an infinite kernel set, the margin is maximized subject to an infinite number of constraints with a compact index set and an additional (Riemann-Stieltjes) integral constraint due to the combinations. After a parametrization in the space of probability measures, we get a semi-infinite programming problem. We analyse regularity conditions (reduction ansatz) and discuss the type of density functions in the constraints and the bilevel optimization problem derived. Our proposed approach is implemented with the conceptual reduction method and tested on homogeneous and heterogeneous data;this yields a better accuracy than a single-kernel learning for the heterogeneous data. We analyse the structure of problems obtained and discuss structural frontiers, trade-offs and research challenges.
An exact estimate on the modulus of metric regularity for linear systems is given. By applying the estimate, we obtain explicit forms of the modulus for linear conical systems and differentiable nonlinear systems on t...
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An exact estimate on the modulus of metric regularity for linear systems is given. By applying the estimate, we obtain explicit forms of the modulus for linear conical systems and differentiable nonlinear systems on the space of continuous functions.
As data become heterogeneous, multiple kernel learning methods may help to classify them. To overcome the drawback lying in its (multiple) finite choice, we propose a novel method of 'infinite' kernel combinat...
详细信息
As data become heterogeneous, multiple kernel learning methods may help to classify them. To overcome the drawback lying in its (multiple) finite choice, we propose a novel method of 'infinite' kernel combinations for learning problems with the help of infinite and semi-infinite optimizations. Looking at all the infinitesimally fine convex combinations of the kernels from an infinite kernel set, the margin is maximized subject to an infinite number of constraints with a compact index set and an additional (Riemann-Stieltjes) integral constraint due to the combinations. After a parametrization in the space of probability measures, we get a semi-infinite programming problem. We analyse regularity conditions (reduction ansatz) and discuss the type of density functions in the constraints and the bilevel optimization problem derived. Our proposed approach is implemented with the conceptual reduction method and tested on homogeneous and heterogeneous data;this yields a better accuracy than a single-kernel learning for the heterogeneous data. We analyse the structure of problems obtained and discuss structural frontiers, trade-offs and research challenges.
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