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===Optimal control===
{{Main
In [[engineering]] and [[economics]], many problems involve multiple objectives which are not describable as the-more-the-better or the-less-the-better; instead, there is an ideal target value for each objective, and the desire is to get as close as possible to the desired value of each objective. For example, energy systems typically have a trade-off between performance and cost<ref>{{Cite journal |last1=Shirazi |first1=Ali |last2=Najafi |first2=Behzad |last3=Aminyavari |first3=Mehdi |last4=Rinaldi |first4=Fabio |last5=Taylor |first5=Robert A. |date=2014-05-01 |title=Thermal–economic–environmental analysis and multi-objective optimization of an ice thermal energy storage system for gas turbine cycle inlet air cooling |journal=Energy |volume=69 |pages=212–226 |doi=10.1016/j.energy.2014.02.071 |hdl=11311/845828 |doi-access=free |bibcode=2014Ene....69..212S |hdl-access=free}}</ref><ref>{{cite journal |last1=Najafi |first1=Behzad |last2=Shirazi |first2=Ali |last3=Aminyavari |first3=Mehdi |last4=Rinaldi |first4=Fabio |last5=Taylor |first5=Robert A. |date=2014-02-03 |title=Exergetic, economic and environmental analyses and multi-objective optimization of an SOFC-gas turbine hybrid cycle coupled with an MSF desalination system |journal=Desalination |volume=334 |issue=1 |pages=46–59 |doi=10.1016/j.desal.2013.11.039 |hdl=11311/764704 |doi-access=free |bibcode=2014Desal.334...46N |hdl-access=free}}</ref> or one might want to adjust a rocket's fuel usage and orientation so that it arrives both at a specified place and at a specified time; or one might want to conduct [[open market operations]] so that both the [[inflation rate]] and the [[unemployment rate]] are as close as possible to their desired values.
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:where the weights of the objectives <math>w_i>0</math> are the parameters of the scalarization. If the parameters/weights are drawn uniformly in the positive orthant, it is shown that this scalarization provably converges to the [[Pareto front]],<ref name="Golovin2021" /> even when the front is non-convex.
=== Smooth Chebyshev (Tchebycheff) scalarization ===
The '''smooth Chebyshev scalarization''';<ref name="Lin2024">Lin, X.; Zhang, X.; Yang, Z.; Liu, F.; Wang, Z.; Zhang, Q. (2024). “Smooth Tchebycheff Scalarization for Multi-Objective Optimization”. ''arXiv preprint'' [[arXiv:2402.19078]].</ref> also called smooth Tchebycheff scalarisation (STCH); replaces the non-differentiable max-operator of the classical Chebyshev scalarization with a smooth logarithmic soft-max, making standard gradient-based optimization applicable. Unlike typical scalarization methods, it guarantees exploration of the entire Pareto front, convex or concave.
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</math>
where <math>u</math> is the ''smoothing parameter'' and <math>\boldsymbol{\lambda}=(\lambda_{1},\dots ,\lambda_{k})</math> is a weight vector on the probability simplex <math>\Delta_{k-1}</math>.
As <math>u\to 0^{+}</math> this converges to the classical (non-smooth) Chebyshev form
<math>
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;Properties
* '''Smoothness and complexity''' — <math>g_{u}^{\mathrm{STCH}}</math> is continuously differentiable with an <math>L</math>-Lipschitz gradient. When every <math>f_{i}</math> is convex the function is convex, and an <math>\varepsilon</math>-optimal point is reachable in <math>\mathcal{O}(1/\varepsilon)</math> first-order iterations; sub-gradient descent on <math>g^{\mathrm{TCH}}</math> needs <math>\mathcal{O}(1/\varepsilon^{2})</math> iterations.<ref name="Lin2024"
* '''Pareto optimality''' — For any <math>u>0</math> every minimizer of <math>g_{u}^{\mathrm{STCH}}(\cdot\mid\boldsymbol{\lambda})</math> is weakly Pareto-optimal; if all <math>\lambda_{i}>0</math> (or the minimizer is unique) it is Pareto-optimal.<ref name="Lin2024"
* '''Exhaustiveness''' — There exists a threshold <math>u^{*}>0</math> such that, for <math>0<u<u^{*}</math>, every Pareto-optimal point can be obtained as a minimizer of <math>g_{u}^{\mathrm{STCH}}</math> for some weight vector <math>\boldsymbol{\lambda}</math>; when the Pareto front is convex this holds for all <math>u>0</math>.<ref name="Lin2024"
For example, [[portfolio optimization]] is often conducted in terms of [[Modern portfolio theory|mean-variance analysis]]. In this context, the efficient set is a subset of the portfolios parametrized by the portfolio mean return <math>\mu_P</math> in the problem of choosing portfolio shares to minimize the portfolio's variance of return <math>\sigma_P</math> subject to a given value of <math>\mu_P</math>; see [[Mutual fund separation theorem#Portfolio separation in mean-variance analysis|Mutual fund separation theorem]] for details. Alternatively, the efficient set can be specified by choosing the portfolio shares to maximize the function <math>\mu_P - b \sigma_P </math>; the set of efficient portfolios consists of the solutions as <math>b</math> ranges from zero to infinity.
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