In this paper, we deal with a new class of SOS-convex (sum of squares convex) polynomialoptimization problems with spectrahedral uncertainty data in both the objective and constraints. By using robust optimization an...
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In this paper, we deal with a new class of SOS-convex (sum of squares convex) polynomialoptimization problems with spectrahedral uncertainty data in both the objective and constraints. By using robust optimization and a weighted-sum scalarization methodology, we first present the relationship between robust solutions of this uncertain SOS-convex polynomial optimization problem and that of its corresponding scalar optimization problem. Then, by using a normal cone constraint qualification condition, we establish necessary and sufficient optimality conditions for robust weakly efficient solutions of this uncertain SOS-convex polynomial optimization problem based on scaled diagonally dominant sums of squares conditions and linear matrix inequalities. Moreover, we introduce a semidefinite programming relaxation problem of its weighted-sum scalar optimization problem, and show that robust weakly efficient solutions of the uncertain SOS-convex polynomial optimization problem can be found by solving the corresponding semidefinite programming relaxation problem.
This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convexpolynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex poly...
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This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convexpolynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem (P-z) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem (P-z), we establish two kinds of representations of non-negativity of convexpolynomials over convex semi-algebraic sets, and propose two kinds of finite convergence results of the Lasserre-type hierarchy of semidefinite programming relaxations for problem (P-z) under suitable assumptions. Finally, we show that finding efficient solutions to problem (MP) can be achieved successfully by solving hierarchies of semidefinite programming relaxations and checking a flat truncation condition.
Probabilistic failure envelopes are commonly used to represent the bearing capacity of skirted foundations in spatially variable soils. Traditional methods rely on probe loading and machine learning (ML) models, but t...
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Probabilistic failure envelopes are commonly used to represent the bearing capacity of skirted foundations in spatially variable soils. Traditional methods rely on probe loading and machine learning (ML) models, but these can lead to error propagation due to sparse data. This paper proposes a novel ML based framework for uncertainty quantification, replacing probe tests with a simplified modified swipe (SMS) method. The SMS method generates complete failure envelopes more quickly and accurately while providing more features to prevent error propagation, reducing errors by 43%. Among five ML methods-multilayer perceptron (MLP), convolutional neural network (CNN), k-nearest neighbour (KNN), decision tree (DT), and support vector machine (SVM)-MLP performed best, with prediction errors under 5% for coefficient of variation (COV) of 0.1 and below 10% for COV = 0.5. Additionally, the convex polynomial optimization method replaces the conventional fitting function and yields closed-form functions, which accurately represent various envelopes in random soil. The framework was validated across five different skirt lengths and soil conditions, providing safety factors for a 1% failure probability based on 4000 samples. This procedure holds promise for aiding in the reliability design of other offshore foundations.
The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J. B. Lasserre, convexity in semialgebraic geometry a...
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The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J. B. Lasserre, convexity in semialgebraic geometry and polynomialoptimization, SIAM J. Optim., 19 (2009), pp. 1995-2014]. We give a new proof of the finite convergence property under weaker assumptions than were known before. In addition, we show that-under the assumptions for finite convergence-the number of steps needed for convergence depends on more than the input size of the problem.
In this paper, under a suitable regularity condition, we establish a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convexpolynomial programs, exhibiting exact conic program...
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In this paper, under a suitable regularity condition, we establish a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convexpolynomial programs, exhibiting exact conic programming relaxation, which can be solved by various numerical methods such as interior point methods. By considering a general convex cone program, we give unified results that apply to many classes of important cone programs, such as the second-order cone programs, semidefinite programs, and polyhedral cone programs. When the cones involved in the programs are polyhedral cones, we present a regularity-free exact semidefinite programming relaxation. We do this by establishing a sum-of-squares polynomial representation of positivity of a real sum-of-squares convexpolynomial over a conic sum-of-squares convex system. In many cases, the sum-of-squares representation can be numerically checked via solving a conic programming problem. Consequently, we also show that a convex set, described by a conic sum-of-squares convexpolynomial, is (lifted) conic linear representable in the sense that it can be expressed as (a projection of) the set of solutions to some conic linear systems.
We show that the Lasserre hierarchy of semidefinite programming (SDP) relaxations with a slightly extended quadratic module for convex polynomial optimization problems always converges asymptotically even in the case ...
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We show that the Lasserre hierarchy of semidefinite programming (SDP) relaxations with a slightly extended quadratic module for convex polynomial optimization problems always converges asymptotically even in the case of non-compact semi-algebraic feasible sets. We then prove that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point guarantees the finite convergence of the hierarchy. We do this by establishing a new sum-of-squares polynomial representation of convexpolynomials over convex semi-algebraic sets. (C) 2013 Elsevier B.V. All rights reserved.
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