An infeasible interior-point algorithm for solving primal and dual geometric programs

“…For example, consider the scalar entropy function This function is smooth over the positive interval, but it is discontinuous at the origin, and its derivative is unbounded near the origin. Both of these features cause problems for some numerical methods [93]. Using the hypograph can solve these problems.…”

Section: Graph Implementationsmentioning

confidence: 99%

Disciplined Convex Programming

Grant

Boyd

Nonconvex Optimization and Its Applications

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3,646

4,391

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“…A huge improvement in computational efficiency was achieved in 1994, when Nesterov and Nemirovsky developed provably efficient interior-point methods for many nonlinear convex optimization problems, including GPs (Nesterov and Nemirovsky 1994). A bit later, Kortanek, Xu, and Ye developed a primal-dual interior-point method for geometric programming, with efficiency approaching that of interior-point linear programming solvers (Kortanek et al 1997). …”

Section: Geometric Programmingmentioning

confidence: 99%

Tractable approximate robust geometric programming

2007

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The optimal solution of a geometric program (GP) can be sensitive to variations in the problem data. Robust geometric programming can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a GP and optimizing for the worst-case scenario under this model. However, it is not known whether a general robust GP can be reformulated as a tractable optimization problem that interior-point or other algorithms can efficiently solve. In this paper we propose an approximation method that seeks a compromise between solution accuracy and computational efficiency.The method is based on approximating the robust GP as a robust linear program (LP), by replacing each nonlinear constraint function with a piecewise-linear (PWL) convex approximation. With a polyhedral or ellipsoidal description of the uncertain data, the resulting robust LP can be formulated as a standard convex optimization problem that interior-point methods can solve. The drawback of this basic method is that the number of terms in the PWL approximations required to obtain an acceptable approximation error can be very large. To overcome the "curse of dimensionality" that arises in directly approximating the nonlinear constraint functions in the original robust GP, we form a conservative approximation of the original robust GP, which contains only bivariate constraint functions. We show how to find globally optimal PWL approximations of these bivariate constraint functions.

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“…The output is related to the input through the relation (11) Differentiating both sides with respect to leads to the familiar result from elementary feedback theory (12) Differentiating again yields (13) and, once more (14) using and from the previous equation. This equation shows that the third-order coefficient of the closed-loop transfer characteristic is given by (15) This is the well-known result showing the linearizing effect of (linear) feedback on an amplifier stage.…”

Section: Nonlinearitymentioning

confidence: 99%

“…A huge improvement in computational efficiency was achieved in 1994, when Nesterov and Nemirovsky developed efficient interior-point algorithms to solve a variety of nonlinear optimization problems, including geometric programs [6]. Recently, Kortanek et al have shown how the most sophisticated primal-dual interior-point methods used in linear programming can be extended to geometric programming, resulting in an algorithm approaching the efficiency of current interior-point linear programming solvers [14]. The algorithm they describe has the desirable feature of exploiting sparsity in the problem, i.e., efficiently handling problems in which each variable appears in only a few constraints.…”

Section: B Solving Geometric Programsmentioning

confidence: 99%

Optimal allocation of local feedback in multistage amplifiers via geometric programming

Dawson

Boyd

Lee

et al.

Proceedings of the 43rd IEEE Midwest Symposium on Circuits and Systems (Cat.No.CH37144)

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Abstract-We consider the problem of optimally allocating local feedback to the stages of a multistage amplifier. The local feedback gains affect many performance indexes for the overall amplifier, such as bandwidth, gain, rise time, delay, output signal swing, linearity, and noise performance, in a complicated and nonlinear fashion, making optimization of the feedback gains a challenging problem. In this paper, we show that this problem, though complicated and nonlinear, can be formulated as a special type of optimization problem called geometric programming. Geometric programs can be solved globally and efficiently using recently developed interior-point methods. Our method, therefore, gives a complete solution to the problem of optimally allocating local feedback gains, taking into account a wide variety of constraints.

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An infeasible interior-point algorithm for solving primal and dual geometric programs

Cited by 111 publications

References 25 publications

Disciplined Convex Programming

Disciplined Convex Programming

Tractable approximate robust geometric programming

Optimal allocation of local feedback in multistage amplifiers via geometric programming

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